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JHEP11(2018)102
Published for SISSA by Springer
Received: September 24, 2018
Accepted: October 22, 2018
Published: November 19, 2018
Light-ray operators in conformal field theory
Petr Kravchuk and David Simmons-Duffin
Walter Burke Institute for Theoretical Physics, Caltech,
Pasadena, California 91125, U.S.A.
E-mail: [email protected], [email protected]
Abstract: We argue that every CFT contains light-ray operators labeled by a continuous
spin J . When J is a positive integer, light-ray operators become integrals of local operators
over a null line. However for non-integer J , light-ray operators are genuinely nonlocal and
give the analytic continuation of CFT data in spin described by Caron-Huot. A key role in
our construction is played by a novel set of intrinsically Lorentzian integral transforms that
generalize the shadow transform. Matrix elements of light-ray operators can be computed
via the integral of a double-commutator against a conformal block. This gives a simple
derivation of Caron-Huot’s Lorentzian OPE inversion formula and lets us generalize it to
arbitrary four-point functions. Furthermore, we show that light-ray operators enter the
Regge limit of CFT correlators, and generalize conformal Regge theory to arbitrary four-
point functions. The average null energy operator is an important example of a light-ray
operator. Using our construction, we find a new proof of the average null energy condition
(ANEC), and furthermore generalize the ANEC to continuous spin.
Keywords: Conformal Field Theory, Field Theories in Higher Dimensions, Conformal
and W Symmetry
ArXiv ePrint: 1805.00098
Open Access, c© The Authors.
Article funded by SCOAP3.https://doi.org/10.1007/JHEP11(2018)102
JHEP11(2018)102
Contents
1 Introduction 1
2 The light transform 7
2.1 Review: Lorentzian cylinder 7
2.1.1 Symmetry between different Poincare patches 9
2.1.2 Causal structure 11
2.2 Review: representation theory of the conformal group 11
2.3 Weyl reflections and integral transforms 14
2.3.1 Transforms for S∆, SJ , S 16
2.3.2 Transform for L 17
2.3.3 Transforms for F,R,R 19
2.4 Some properties of the light transform 20
2.5 Light transform of a Wightman function 24
2.6 Light transform of a time-ordered correlator 26
2.7 Algebra of integral transforms 27
3 Light-ray operators 30
3.1 Euclidean partial waves 30
3.2 Wick-rotation to Lorentzian signature 32
3.3 The light transform and analytic continuation in spin 32
3.3.1 More on even vs. odd spin 35
3.4 Light-ray operators in Mean Field Theory 37
3.4.1 Subleading families and multi-twist operators 40
4 Lorentzian inversion formulae 42
4.1 Inversion for the scalar-scalar OPE 42
4.1.1 The double commutator 42
4.1.2 Inversion for a four-point function of primaries 44
4.1.3 Writing in terms of cross-ratios 47
4.1.4 A natural formula for the Lorentzian block 48
4.2 Generalization to arbitrary representations 50
4.2.1 The light transform of a partial wave 50
4.2.2 The generalized Lorentzian inversion formula 51
4.2.3 Proof using weight-shifting operators 52
5 Conformal Regge theory 54
5.1 Review: Regge kinematics 54
5.2 Review: Sommerfeld-Watson resummation 57
5.3 Relation to light-ray operators 59
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JHEP11(2018)102
6 Positivity and the ANEC 61
6.1 Rindler positivity 62
6.2 The continuous-spin ANEC 63
6.3 Example: Mean Field Theory 66
6.4 Relaxing the conditions on ∆φ 67
7 Discussion 68
A Correlators and tensor structures with continuous spin 71
A.1 Analyticity properties of Wightman functions 71
A.2 Two- and three-point functions 73
A.3 Conventions for two- and three-point tensor structures 76
B Relations between integral transforms 77
B.1 Square of light transform 77
B.2 Relation between shadow transform and light transform 79
C Harmonic analysis for the Euclidean conformal group 80
C.1 Pairings between three-point structures 80
C.2 Euclidean conformal integrals 81
C.3 Residues of Euclidean partial waves 82
D Computation of R(∆1,∆2, J) 84
E Parings of continuous-spin structures 84
E.1 Two-point functions 85
E.2 Three-point pairings 86
F Integral transforms, weight-shifting operators and integration by parts 87
F.1 Euclidean signature 87
F.2 Lorentzian signature 88
G Proof of (4.48) for seed blocks 90
H Conformal blocks with continuous spin 95
H.1 Gluing three-point structures 95
H.1.1 Example: integer spin in Euclidean signature 95
H.1.2 Example: continuous spin in Lorentzian signature 96
H.1.3 Rules for weight-shifting operators 98
H.2 A Lorentzian integral for a conformal block 99
H.2.1 Shadow transform in the diamond 100
H.3 Conformal blocks at large J 102
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JHEP11(2018)102
1
24
3
Figure 1. The Regge limit of a four-point function: the points x1, . . . , x4 approach null infinity,
with the pairs x1, x2 and x3, x4 becoming nearly lightlike separated.
1 Introduction
Singularities of Euclidean correlators in conformal field theory (CFT) are described by
the operator product expansion (OPE). However, in Lorentzian signature there exist sin-
gularities that cannot be described in a simple way using the OPE. One of the most
important is the Regge limit of a time-ordered four-point function (figure 1) [1–6].1 The
Regge limit is the CFT version of a high-energy scattering process: operators O1(x1) and
O3(x3) create excitations that move along nearly lightlike trajectories, interact, and then
are measured by operators O2(x2) and O4(x4). In holographic theories, the Regge limit is
dual to high-energy forward scattering in the bulk [8].
In Lorentzian signature, the OPE Oi ×Oj converges if the product OiOj acts on the
vacuum (either past or future) [9]. That is, we have an equality of states
OiOj |Ω〉 =∑k
fijkOk|Ω〉, (1.1)
where k runs over local operators of the theory (we suppress position dependence, for
brevity). Thus, in figure 1 the OPE O1×O3 converges because it acts on the past vacuum,
the OPE O2×O4 converges because it acts on the future vacuum, and the OPEs O1×O4
and O2×O3 converge because they act on either the past or future vacuum. (Here we use
the fact that spacelike-separated operators commute to rearrange the operators in the time-
ordered correlator to apply (1.1).) However, each of these OPEs is converging very slowly
in the Regge limit. They can be used to prove results like analyticity and boundedness in
the Regge limit [10, 11], but they are less useful for computations (unless one has good
control over the theory). Meanwhile, the OPEs O1 × O2 and O3 × O4 are invalid in the
Regge regime.
1In perturbation theory, Lorentzian singularities correspond to Landau diagrams [7]. It is possible that
this is also true nonperturbatively.
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JHEP11(2018)102
The problem of describing four-point functions in the Regge regime was partially solved
in [3, 8, 12]. The behavior of the correlator is controlled by the analytic continuation of
data in the O1 × O2 and O3 × O4 OPEs to non-integer spin. For example, in a planar
theory, the Regge correlator behaves (very) schematically as
〈O1O2O3O4〉〈O1O2〉〈O3O4〉
∼ 1− f12O(J0)f34O(J0)et(J0−1) + . . . . (1.2)
Here, f12O(J) and f34O(J) are OPE coefficients that have been analytically continued in
the spin J of O. The parameter t measures the boost of O1,O2 relative to O3,O4. J0 ∈ Ris the Regge/Pomeron intercept, and is determined by the analytic continuation of the
dimension ∆O to non-integer J .2 The “. . . ” in (1.2) represent higher-order corrections in
1/N2 and also terms that grow slower than et(J0−1) in the Regge limit t→∞.
A missing link in this story was provided recently by Caron-Huot, who proved that OPE
coefficients and dimensions have a natural analytic continuation in spin in any CFT [16].
The analytic continuation of OPE data in a scalar four-point function 〈φ1φ2φ3φ4〉 can
be computed by a “Lorentzian inversion formula,” given by the integral of a double-
commutator 〈[φ4, φ1][φ2, φ3]〉 times a conformal block GJ+d−1,∆−d+1 with unusual quantum
numbers. Specifically, ∆, J are replaced with
(∆, J)→ (J + d− 1,∆− d+ 1) (1.3)
relative to a conventional conformal block. Caron-Huot’s Lorentzian inversion formula
has many other useful applications, for example to large-spin perturbation theory and the
lightcone bootstrap [17–26], and to the SYK model [27–30].3
However, Caron-Huot’s result raises some obvious questions:
• Can operators themselves (not just their OPE data) be analytically continued in spin?
• What is the space of continuous spin operators in a given CFT?
• Do continuous-spin operators have a Hilbert space interpretation (similar to how
integer-spin operators correspond to CFT states on Sd−1)?
• What is the meaning of the funny block in the Lorentzian inversion formula, and how
do we generalize it?
Answering these questions is important for making sense of the Regge limit, and more
generally for understanding how to write a convergent OPE in non-vacuum states.
It is easy to describe continuous-spin operators mathematically. Consider first a pri-
mary operator Oµ1···µJ (x) with integer spin J . Let us introduce a null polarization vector
zµ and contract it with the indices of O to form a function of (x, z):
O(x, z) ≡ Oµ1···µJ (x)zµ1 · · · zµJ , (z2 = 0). (1.4)
2In d = 2, the Regge regime is the same as the chaos regime. In d ≥ 3, it is related to chaos in
hyperbolic space. See [13, 14] for discussions. Note that J0 − 1 plays the role of a Lyapunov exponent, and
it is constrained by the chaos bound to be less than 1 [10, 15].3In the 1-dimensional SYK model, the analog of analytic continuation in spin is analytic continuation
in the weight of discrete states in the conformal partial wave expansion [29, 31].
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JHEP11(2018)102
The tensor Oµ1···µJ (x) can be recovered from the function O(x, z) by stripping off the
z’s and subtracting traces. Thus, O(x, z) is a valid alternative description of a traceless
symmetric tensor. Note that O(x, z) is a homogeneous polynomial of degree J in z. The
generalization to a continuous spin operator O is now straightforward: we simply drop the
requirement that O(x, z) be polynomial in z and allow it to have non-integer homogeneity,
O(x, λz) = λJO(x, z), λ > 0, J ∈ C. (1.5)
Continuous-spin operators are necessarily nonlocal. This follows from Mack’s classifi-
cation of positive-energy representations of the Lorentzian conformal group SO(d, 2) [32],
which only includes nonnegative integer spin representations.4 CFT states have positive
energy, so by the state-operator correspondence, local operators must have nonnegative
integer spin, and conversely continuous-spin operators must be nonlocal. Mack’s classifi-
cation also shows that continuous-spin operators must annihilate the vacuum:
O(x, z)|Ω〉 = 0 (J /∈ Z≥0), (1.6)
otherwise O(x, z)|Ω〉 would transform in a nontrivial continuous-spin representation, which
would include a state with negative energy.
If continuous-spin operators annihilate the vacuum, how can we analytically continue
the local operators of a CFT, which certainly do not annihilate the vacuum? The answer is
that we must first turn local operators into something nonlocal that annihilates the vacuum,
and then analytically continue that. The correct object turns out to be the integral of a
local operator along a null line,∫ ∞−∞
dαO(αz, z) =
∫ ∞−∞
dαOµ1···µJ (αz)zµ1 · · · zµJ . (1.7)
This can be written more covariantly by performing a conformal transformation to bring
the beginning of the null line to a generic point x:5
L[O](x, z) ≡∫ ∞−∞
dα(−α)−∆−JO(x− z
α, z). (1.8)
This defines an integral transform L that we call the “light transform.” The expression (1.7)
corresponds to L[O](−∞z, z), where x = −∞z is a point at past null infinity.
After reviewing some representation theory in sections 2.1 and 2.2, we show in sec-
tion 2.3 that if O∆,J has dimension ∆ and spin J , then L[O∆,J ](x, z) transforms like a
primary operator with dimension 1 − J and spin 1−∆:
L : (∆, J)→ (1− J, 1−∆). (1.9)
4For non traceless-symmetric tensor operators, we define spin as the length of the first row of the Young
diagram for their SO(d) representation. For fermionic representations spin is a half-integer and for simplicity
of language we include this case into the notion of “integer spin” operators.5As α → 0−, the point x − z/α diverges to future null infinity, and the integration contour should be
understood as extending into the next Poincare patch on the Lorentzian cylinder. We give more detail in
section 2.3.2.
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JHEP11(2018)102
In particular, L[O∆,J ] can have non-integer spin. The average null energy operator E =
L[T ] (the light transform of the stress tensor) is a special case, having dimension −1 and
spin 1−d. We will see that L is part of a dihedral group (D8) of intrinsically Lorentzian inte-
gral transforms that generalize the Euclidean shadow transform [33, 34]. These Lorentzian
transforms implement affine Weyl reflections that preserve the Casimirs of the conformal
group. For example, the quadratic Casimir eigenvalue is given by
C2(∆, J) = ∆(∆− d) + J(J + d− 2), (1.10)
and this is indeed invariant under (1.9). The transformation (1.3) appearing in Caron-
Huot’s formula is another affine Weyl reflection. The Lorentzian transforms do not give
precisely a representation of D8, but instead satisfy an interesting “anomalous” algebra
that we derive in section 2.7. Mack’s classification implies that L[O∆,J ] must annihilate
the vacuum whenever O∆,J is a local operator. This is also easy to see directly by deforming
the α contour into the complex plane, as we show in section 2.4.
We claim that the operators L[O∆,J ] can be analytically continued in J , and their
continuations are light-ray operators.6 As an example, consider Mean Field Theory (a.k.a.
Generalized Free Fields) in d = 2 with a scalar primary φ. This theory contains “double-
trace” operators
[φφ]J(u, v) ≡:φ(u, v)∂Jv φ(u, v) : + ∂v(. . .) (1.11)
with dimension 2∆φ + J and even spin J . Here, : : denotes normal ordering and we have
written out the definition up to total derivatives (which are required to ensure that this
is a primary operator). We are using lightcone coordinates u = x − t, v = x + t, and for
simplicity focusing on operators with ∂v derivatives only. The corresponding analytically-
continued light-ray operators are
OJ(0,−∞) =iΓ(J+1)
2J
∫ ∞−∞
dv
∫ ∞−∞
ds
2π
(1
(s+iε)J+1+
1
(−s+iε)J+1
):φ(0,v+s)φ(0,v−s): .
(1.12)
When J is an even integer, we have
iΓ(J + 1)
2π
(1
(s+ iε)J+1− 1
(s− iε)J+1
)=∂Jδ(s)
∂sJ(J ∈ 2Z≥0). (1.13)
Thus, when J is an even integer, OJ becomes
OJ(0,−∞) = 2−J∫ ∞−∞
dv
∫ ∞−∞
ds∂Jδ(s)
∂sJ:φ(0, v + s)φ(0, v − s) :
=
∫ ∞−∞
dv :φ∂Jv φ : (0, v) = L[[φφ]J ](0,−∞) (J ∈ 2Z≥0). (1.14)
6Note that L[O∆,J ](x, z) has dimension 1− J and spin 1−∆. Thus, analytic continuation in J is really
analytic continuation in the dimension of L[O∆,J ] away from negative integer values. We will continue to
refer to it as analytic continuation in spin, since J labels the spin of local operators.
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JHEP11(2018)102
By contrast, when J is not an even integer, OJ is a legitimately nonlocal light-ray operator
whose correlators are analytic continuations of the correlators of L[[φφ]J ]. In particular,
three-point functions 〈O1O2OJ〉 give an analytic continuation of the three-point coefficients
of 〈O1O2[φφ]J〉.Similar light-ray operators have a long history in the gauge-theory literature [35, 36]
(see [37–40] for recent discussions). There, one often considers a bilocal integral of operators
inserted along a null Wilson line. Such operators were discussed in [41], where they were
argued to control OPEs of the average null energy operator E . In perturbation theory, it
is reasonable to imagine constructing more operators like (1.12). However, it is less clear
how to define them in a nonperturbative context where normal ordering is not well-defined,
and there can be complicated singularities when two operators become lightlike-separated.
It is also not clear what a null Wilson line means in an abstract CFT.
Our tool for constructing analogs of OJ in general CFTs will be harmonic analysis [42].
Given primary operators O1,O2, we find in section 3 an integration kernel K∆,J(x1, x2, x, z)
such that
O∆,J(x, z) =
∫ddx1d
dx2K∆,J(x1, x2, x, z)O1(x1)O2(x2) (1.15)
transforms like a primary with dimension 1 − J and spin 1−∆ (when inserted in a time-
ordered correlator). The object O∆,J is meromorphic in ∆ and J and has poles of the form
O∆,J(x, z) ∼ 1
∆−∆i(J)Oi,J(x, z). (1.16)
We conjecture based on examples that poles must come from the region where x1, x2 are
close to the light ray x+R≥0z (we have not established this rigorously in a general CFT).
The residues of the poles can thus be interpreted as light-ray operators Oi,J(x, z) that
make sense in arbitrary correlators. Furthermore, when J is an integer, the residues are
light-transforms of local operators L[O]. Thus the Oi,J give analytic continuations of L[O]
for all O ∈ O1 ×O2.
In section 4, we show that 〈O3O4O∆,J〉 can be computed via the integral of a double-
commutator 〈[O4,O1][O2,O3]〉 over a Lorentzian region of spacetime. This leads to a simple
proof of Caron-Huot’s Lorentzian inversion formula. The contour manipulation from [31]
is crucial for this computation. However, the light-ray perspective makes our proof simpler
than the one in [31]. In particular, it makes it clearer why the unusual conformal block
GJ+d−1,∆−d+1 appears. The reason is that the quantum numbers (J + d − 1,∆ − d + 1)
are dual to those of the light-transform (1 − J, 1−∆) in the sense that the product
ddx ddz δ(z2)O1−J,1−∆(x, z)OJ+d−1,∆−d+1(x, z) (1.17)
has dimension zero and spin zero. Our perspective also leads to a natural generalization of
Caron-Huot’s formula to the case of arbitrary operator representations, which we describe
in section 4.2. Subsequently in section 5, we generalize conformal Regge theory to arbitrary
operator representations as well, along the way showing that light-ray operators describe
part of the Regge limit of four-point functions as conjectured in [6].
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JHEP11(2018)102
As mentioned above, the average null energy operator E = L[T ] is an example of a
light-ray operator. The average null energy condition (ANEC) states that E is positive-
semidefinite, i.e. its expectation value in any state is nonnegative. Some implications of
the ANEC in CFTs are discussed in [41, 43, 44]. The ANEC was recently proven in [45]
using techniques from information theory and in [46] using causality. By expressing E as
the residue of an integral of a pair of real operators φ(x1)φ(x2), we find a new proof of the
ANEC in section 6.7 Furthermore, E is part of a family of light-ray operators EJ labeled by
continuous spin J , and our construction of light-ray operators applies to this entire family.
This lets us derive a novel generalization of the ANEC to continuous spin. More precisely,
we show that
〈Ψ|EJ |Ψ〉 ≥ 0, (J ∈ R≥Jmin), (1.18)
where EJ is the family of light-ray operators whose values at even integer J are given by
EJ = L[O∆min(J),J ] (J ∈ 2Z, J ≥ 2), (1.19)
where O∆min(J),J is the operator with spin J of minimal dimension. Here, Jmin ≤ 1 is the
smallest value of J for which the Lorentzian inversion formula holds [16].
We conclude in section 7 with discussion and numerous questions for the future. The
appendices contain useful mathematical background, further technical details, and some
computations needed in the main text. In particular, appendix A includes a general dis-
cussion of continuous-spin tensor structures and their analyticity properties, appendix C
contains a lightning review of harmonic analysis for the Euclidean conformal group, and
appendix H gives details on conformal blocks with continuous spin.
Notation. In this work, we use the convention that correlators in the state |Ω〉 represent
physical correlators in a CFT. For example,
〈Ω|O1 · · · On|Ω〉 (1.20)
is a physical Wightman function, and
〈O1 · · · On〉Ω ≡ 〈Ω|TO1 · · · On|Ω〉 (1.21)
is a physical time-ordered correlator.
Often, we discuss two- and three-point structures that are fixed by conformal invariance
up to a constant. These structures do not represent physical correlators — they are simply
known functions of spacetime points. We write them as correlators in the ficticious state
|0〉. For example, if φi are scalar primaries with dimensions ∆i, then
〈0|φ1(x1)φ2(x2)φ3(x3)|0〉= 1
(x212+iεt12)
∆1+∆2−∆32 (x2
23+iεt23)∆2+∆3−∆1
2 (x213+iεt13)
∆1+∆3−∆22
(1.22)
7Our proof is conceptually very similar to the one in [46], but it has a technical advantage that it does
not require any assumptions about the behavior of correlators outside the regime of OPE convergence. A
disadvantage is that we require the dimension ∆φ to be sufficiently low, though we expect it should be
possible to relax this restriction.
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JHEP11(2018)102
denotes the unique conformally-invariant three-point structure for scalars with dimensions
∆i, with the iε-prescription appropriate for the given Wightman ordering. Similarly,
〈φ1(x1)φ2(x2)φ3(x3)〉 =1
(x212 + iε)
∆1+∆2−∆32 (x2
23 + iε)∆2+∆3−∆1
2 (x213 + iε)
∆1+∆3−∆22
(1.23)
denotes the unique conformally-invariant structure with the iε-prescription for a time-
ordered correlator. In particular, (1.22) and (1.23) do not include OPE coefficients.
2 The light transform
This section is devoted to mathematical background and results that will be needed for
constructing and studying light-ray operators. We first review some basic facts about the
Lorentzian conformal group and its representation theory, with an emphasis on continuous
spin operators. We then introduce a set of intrinsically Lorentzian integral transforms,
which generalize the well-known Euclidean shadow transform, and study their properties.
One of these transforms is the “light transform” mentioned in the introduction. It will play
a key role in the sections that follow.
2.1 Review: Lorentzian cylinder
Similarly to Euclidean space Rd, Minkowski spaceMd = Rd−1,1 is not invariant under finite
conformal transformations. In Euclidean space, this problem is easily solved by studying
CFTs on Sd, the conformal compactification of Rd. In Lorentzian signature, the problem
is more subtle.
The simplest extension of Minkowski space Md = Rd−1,1 that is invariant under the
Lorentzian conformal group SO(d, 2) is its conformal compactification Mcd. The space
Mcd can be easily described by the embedding space construction [5, 47–52]: it is the
projectivization of the null cone in Rd,2 on which SO(d, 2) acts by its vector representation.
If we choose coordinates on Rd,2 to be X−1, X0, . . . Xd with the metric
X2 = −(X−1)2 − (X0)2 + (X1)2 + . . .+ (Xd)2, (2.1)
then the null cone is defined by
(X−1)2 + (X0)2 = (X1)2 + . . .+ (Xd)2. (2.2)
If we mod out by positive rescalings (i.e. by R+), we can set both sides of this equation to
1, identifying the space of solutions with S1 × Sd−1, where the S1 is timelike. To get Mcd,
we mod out by R rescalings,8 obtainingMcd = S1×Sd−1/Z2, where Z2 identifies antipodal
points in both S1 and Sd−1. Minkowski space Md ⊂ Mcd can be obtained by introducing
lightcone coordinates in Rd,2,
X± = X−1 ±Xd, (2.3)
8In the Euclidean embedding space construction based on Rd+1,1 we usually just take the future null cone
instead of considering negative rescalings, but in Rd,2 the null cone is connected and this is not possible.
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JHEP11(2018)102
∞ ∞Md
Md
Figure 2. Poincare patch Md (blue, shaded) inside the Lorentzian cylinder Md in the case of 2
dimensions. The spacelike infinity of Md is marked by ∞. The dashed lines should be identified.
and considering points with X+ 6= 0. Using R rescalings we can set X+ = 1 for such
points, and the null cone equation becomes
X− = −(X0)2 + (X1)2 + . . .+ (Xd−1)2. (2.4)
If we set xµ = Xµ for µ = 0, . . . d− 1, this gives the standard embedding of Rd−1,1,
(X+, X−, Xµ) = (1, x2, xµ). (2.5)
One can check that the action of SO(d, 2) on X induces the usual conformal group action
on xµ. The points that lie in Mcd\Md have X+ = 0 and thus XµXµ = 0 with arbitrary
X−. They correspond to space-time infinity9 (Xµ = 0) and null infinity (Xµ 6= 0).
By construction, Mcd has an action of SO(d, 2) and is thus a natural candidate for the
space on which a conformally-invariant QFT can live. However, it is unsuitable for this
purpose due to the existence of closed timelike curves that are evident from its description
as S1 × Sd−1/Z2 with timelike S1. This problem can be fixed by instead considering the
universal cover Md = R × Sd−1,10 which is simply the Lorentzian cylinder. It was shown
in [53] that Wightman functions of a CFT on Rd−1,1 can be analytically continued to Md.
Indeed, one can first Wick-rotate the CFT to Rd, map it conformally to the Euclidean
cylinder R × Sd−1, and then Wick-rotate to Md (of course the actual proof in [53] is
more involved).
To describe coordinates on Md, it is convenient to first consider the null cone in Rd,2
mod R+. It is equivalent to S1 × Sd−1 defined by
(X−1)2 + (X0)2 = (X1)2 + . . .+ (Xd)2 = 1, (2.6)
9In Mcd the infinite future, the infinite past and the spatial infinity of Minkowski space are identified.
The past neighborhood of the future infinity, the future neighborhood of the past infinity and the spacelike
neighborhood of the spatial infinity together form a complete neighbourhood of the space-time infinity
in Mcd.
10For d = 2 this is not the universal cover.
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JHEP11(2018)102
and we can use the parametrization
X−1 = cos τ,
X0 = sin τ,
Xi = ei, i = 1 . . . d, (2.7)
where ~e is a unit vector in Rd. Here τ is the coordinate on S1 with identification τ ∼ τ+2π,
and taking the universal cover is equivalent to removing this identification. The coordinates
(τ,~e) with τ ∈ R then cover Md completely. Minkowski space Md can be conformally
identified with a particular region in Md by using the embedding (2.5). This gives
x0 =sin τ
cos τ + ed,
xi =ei
cos τ + ed, i = 1, . . . d− 1, (2.8)
in the region where cos τ +ed > 0 and −π < τ < π. This region consists of points spacelike
separated from τ = 0, ~e = (0, . . . , 0,−1), which is the spatial infinity of Md (see figure 2).
We will refer to this particular region as the (first) Poincare patch. Note that the null
cone in Rd,2 modulo R+ contains two Poincare patches — one with X+ > 0 and one with
X+ < 0. The relation between Wightman functions on Md and Md (in their natural
metrics) for operators reads as11
〈Ω|O1(x1) · · · On(xn)|Ω〉Md=
n∏i=1
(cos τi + edi )∆i〈Ω|O1(τ1, ~e1) · · · On(τn, ~en)|Ω〉Md
. (2.9)
Let us discuss the action of the conformal group on Md. First of all, because we have
taken the universal cover of Mcd, it is no longer true that SO(d, 2) acts on Md. Instead,
the universal covering group SO(d, 2) acts on Md. Indeed, the rotation generator M−1,0
generates shifts in τ and in SO(d, 2) we have e2πM−1,0 = 1, whereas this is definitely not
true on Md because τ τ + 2π. In the universal cover SO(d, 2), this direction gets
decompactified so that the action becomes consistent.
2.1.1 Symmetry between different Poincare patches
There exists an important symmetry T of Md that commutes with the action of SO(d, 2).
Namely, if we take a point with coordinates p = (τ,~e) and send light rays in all future
directions, they will all converge at the point T p ≡ (τ + π,−~e). The points p and T p in
Md correspond to the same point inMcd and thus T commutes with infinitesimal conformal
generators and therefore also with the full SO(d, 2).
When d is even, T lies in the center of SO(d, 2) and we can take
T = eπM−1,0eπM1,2+πM3,4+...+πMd−1,d . (2.10)
11When applied to operators with spin, this identity does not produce a nice function on Md, because in
typical bases of spin indices on Minkowski space translations in τ act by matrices which have singularities.
Therefore, in order to have nice functions on Md one has to perform a redefinition of spin indices [53].
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JHEP11(2018)102
For odd d only T 2 lies in SO(d, 2). But if the theory preserves parity, i.e. we have an
operator P that maps x1 → −x1 in the first Poincare patch, then we can take
T = eπM0,−1+πM23+...+πMd−1,dP. (2.11)
If the theory doesn’t preserve parity, T can still be defined as an operation on correlation
functions in the sense specified below.
If T exists as a unitary operator on the Hilbert space (d even or parity-preserving theory
in odd d), then we can consider its action on local operators. For scalars we clearly have
T φ(x)T −1 = φ(T x), (2.12)
up to intrinsic parity in odd d. To understand the action of T on operators with spin, it is
convenient to work in the embedding space, where we have for tensor operators
T O(X,Z1, Z2, . . . Zn)T −1 = O(−X,−Z1,−Z2, . . . ,−Zn). (2.13)
Here the point −X is interpreted as the point in the Poincare patch which is in immediate
future of the first Poincare patch, and Zi are null polarizations corresponding to the various
rows of the Young diagram of O. Again, in odd dimensions we might need to add a factor
of intrinsic parity.
Note that the above action on tensor operators can be defined regardless of the dimen-
sion d or whether or not the theory preserves parity. We will thus define T as an operator
which can act on functions on Md according to
(T · O)(X,Z1, Z2, . . . Zn) ≡ O(−X,−Z1,−Z2, . . . ,−Zn), (2.14)
where again −X is interpreted as corresponding to T x. As discussed above, in even
dimensions this always comes from a unitary symmetry of the theory defined by (2.10), but
in odd dimensions it may not be a symmetry (even if the theory preserves parity). In such
cases we can still use T thus defined to study conformally-invariant objects, similarly to
how we can separate tensor structures into parity-odd and parity-even regardless of whether
the theory preserves parity. To have a uniform discussion, we will use this definition of Taction in the rest of the paper.
Finally, let us note that in even dimensions for tensor operators
T O(x)|Ω〉 = eiπ(∆+N)O(x)|Ω〉,
〈Ω|O(x)T = eiπ(∆+N)〈Ω|O(x), (2.15)
where N is the total number of boxes in the SO(d − 1, 1) Young diagram of O. This
follows from the fact that the representation generated by O acting on the vacuum is
irreducible. One can check the eigenvalue by considering this identity inside a Wightman
two-point function. The same relation holds in parity-even structures in odd dimensions
(in particular, in two-point functions) and with a minus sign in parity-odd structures.
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JHEP11(2018)102
12
2+
2−
Figure 3. 1 is spacelike from 2 (1 ≈ 2) if and only if 1 is in the future of 2− and the past of 2+
(2− < 1 < 2+). The figure shows the Lorentzian cylinder in 2-dimensions. The dashed lines should
be identified.
2.1.2 Causal structure
The action of SO(d, 2) on Md preserves the causal structure of the Lorentzian cylinder [53].
This property will allow us to define conformally-invariant integration regions. We usually
label points in Md by natural numbers and we write 1 < 2 when point 1 is inside the past
lightcone of 2 and 1 ≈ 2 when 1 is spacelike from 2. Furthermore, we write 1± for T ±11
(more generally, 1±k for T ±k1). That is, 1+ is the point in the “next” Poincare patch with
the same Minkowski coordinates as 1. Similarly, 1− is the point in the “previous” Poincare
patch with the same Minkowski coordinates as 1. Some causal relationships between points
can be written in different ways, for example 1 ≈ 2 if and only if 2− < 1 < 2+ (figure 3).
2.2 Review: representation theory of the conformal group
We will also need some facts from unitary representation theory of the conformal groups
SO(d + 1, 1) and SO(d, 2). These groups are non-compact and their unitary representa-
tions are infinite-dimensional. We will mostly be interested in a particular class of unitary
representations known as principal series representations, and also their non-unitary ana-
lytic continuations.
Unitary principal series representations of SO(d+ 1, 1) are the easiest to describe. In
this case, a principal series representation E∆,ρ is labeled by a pair (∆, ρ), where ∆ is a
scaling dimension of the form ∆ = d2 + is with s ∈ R and an ρ is an irreducible SO(d)
representation. The elements of E∆,ρ are functions on Rd (more precisely, on the conformal
sphere Sd) that transform under SO(d+ 1, 1) as primary operators with scaling dimension
∆ and SO(d) representation ρ. The inner product between two functions fa(x) and ga(x)
(where a is an index for ρ) is defined by
(f, g) ≡∫ddx(fa(x))∗ga(x). (2.16)
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JHEP11(2018)102
This is positive-definite by construction. It is conformally-invariant because while g trans-
forms with scaling dimension ∆ = d2 + is in ρ of SO(d), f∗ transforms with scaling dimen-
sion ∆∗ = d2 − is in ρ∗ of SO(d), and thus the integrand is a scalar of scaling dimension
∆ + ∆∗ = d, as required for conformal invariance. The representations E∆,ρ are important
because the representations of primary operators that appear in CFTs are their analytic
continuations to real ∆.12 Also, E∆,ρ appear in partial wave analysis of Euclidean correla-
tors [42].
The pair (∆, ρ) can be thought of as a weight of the algebra soC(d + 2) if we define
−∆ to be the length of the first row of a Young diagram, and use the Young diagram
of ρ for the remaining rows. Through this identification, the unitary representations of
SO(d+ 2) have non-positive (half-)integer ∆. For SO(d+ 1, 1), we instead have continuous
∆ because the corresponding Cartan generator D ∝M−1,d+1 of SO(d+1, 1) is noncompact
(i.e. it must be multiplied by i in order to relate the Lie algebra so(d+1, 1) to the compact
form so(d+ 2)).
In SO(d, 2) there are two noncompact Cartan generators (D and M01), and both of
their weights become continuous. Thus, the unitary principal series representations P∆,J,λ
for SO(d, 2) are parametrized by a triplet (∆, J, λ), where ∆ ∈ d2 + iR, J ∈ −d−2
2 + iR and
λ is an irrep of SO(d − 2). Here the pair (J, λ) can be thought of as a weight of SO(d),
where J is the component corresponding to the length of the first row of a Young diagram.
In this sense we have a continuous-spin generalization of SO(d) irreps.
To make sense of functions with continuous spin, we follow the logic described in the
introduction. Let us first review the case of integer spin, and take λ to be trivial for
simplicity. The elements of integer spin representations are tensors that are traceless and
symmetric in their indices
fµ1···µJ (x). (2.17)
We can always contract f with a null polarization vector zµ to obtain a homogeneous
polynomial of degree J in z,
f(x, z) ≡ fµ1···µJ (x)zµ1 · · · zµJ . (2.18)
The tensor fµ1···µJ (x) can be recovered from f(x, z) via
fµ1···µJ (x) =1
J !(d−22 )J
Dµ1 · · ·DµJf(x, z), (2.19)
where
Dµ =
(d− 2
2+ z · ∂
∂z
)∂
∂zµ− 1
2zµ
∂2
∂z2(2.20)
is the Thomas/Todorov operator [54–56]. Thus, the two ways (2.17) and (2.18) of repre-
senting f are equivalent.
12It will not be important to give a precise meaning to this “analytic continuation”; in most of the
paper we only use E∆,ρ as a guide for writing conformally-invariant formulas. The same remark concerns
representations of SO(d, 2) below.
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JHEP11(2018)102
The generalization to continuous spin is now as stated in the introduction: we can
consider functions f(x, z) that are homogeneous of degree J in z, where J is no longer an
integer and f(x, z) is no longer a polynomial in z. More precisely, the elements of P∆,J
are functions f(x, z) with x ∈ Mcd and z ∈ Rd−1,1
+ a future-pointing null vector that are
constrained to satisfy
f(x, αz) = αJf(x, z), α > 0. (2.21)
The object f(x, z) transforms under conformal transformations in the same way as func-
tions of the form (2.18) would. The operation of recovering the underlying tensor (2.19)
only makes sense when J is a nonnegative integer.13
To describe representations P∆,J,λ with non-trivial λ, we can make use of an analogy
between the space of polarization vectors z and the embedding space. The embedding
space lets us lift functions on Rd with indices for an SO(d) representation to functions on
the null cone in d + 2 dimensions with indices for an SO(d + 1, 1) representation. In the
present case, λ is a representation of SO(d − 2), so we can lift it to a representation of
SO(d− 1, 1) defined on the null cone z2 = 0 in a similar way. For example, if λ is a rank-k
tensor representation of SO(d− 2), then we consider functions
fa1...ak(x, z), (2.22)
with ai being SO(d − 1, 1)-indices, where f obeys gauge redundancies and transverseness
constraints [58]
fa1...ak(x, z) ∼ fa1...ak(x, z) + zaiha1...ai−1ai+1...ak(x, z), (2.23)
zaifa1...ak(x, z) = 0. (2.24)
Additionally, f should be homogeneous (2.21) and satisfy the same tracelessness and sym-
metry conditions in ai as λ-tensors of SO(d− 2).14 Other types of representations can be
described by adapting other embedding space formalisms. In most of this paper we focus
on trivial λ for simplicity.
We can define an inner product for Lorentzian principal series representations by
(f, g) ≡∫ddxDd−2zf∗(x, z)g(x, z), (2.26)
Dd−2z ≡ ddzθ(z0)δ(z2)
volR+. (2.27)
13Also, f(x, z) should satisfy a differential equation in z. This differential equation is conformally invariant
and is essentially a generalization of the (d − 2)-dimensional conformal Killing equation, similarly to the
equations discussed in [57]. Such equations only exist for nonnegative integer J and express the fact that
f(x, z) is actually polynomial in z.14To make more direct contact with integer spin, instead of (2.23) one can use
Daifa1...ak (x, z) = 0, (2.25)
where D is the Todorov operator acting on z. In this case, for integer spin tensors the function fa1...ak (x, z)
is given simply by contracting zµ with the first-row indices of the tensor.
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JHEP11(2018)102
Here the integral over z replaces the index contraction that we would use for integer J . The
measure for z is manifestly Lorentz-invariant and supported on the null cone. Together
with the measure, the integrand is invariant under rescaling of z. Thus, we obtain a finite
result by dividing by the volume of the group of positive rescalings, volR+. The z-integral
is exactly the kind of integral considered in [34] in the context of the embedding space
formalism. Here, we have adapted it to describe SO(d− 1, 1)-invariant integration on the
null cone z2 = 0.
In section 2.3 we will use analytic continuations of P∆,J,λ to find interesting relations for
primary operators in Lorentzian CFTs. But before we can do this, we should note that these
representations are constructed on Mcd, which is unsatisfactory from the physical point of
view. We can construct similar representations of SO(d, 2) consisting of functions on Md,
which we call P∆,J,λ. These representations behave very similarly to P∆,J,λ but there is an
important distinction. While the representations P∆,J,λ are generically irreducible, their
analogues P∆,J,λ are not. Indeed, the action of T on Md commutes with the action of
SO(d, 2) and thus P∆,J,λ decompose into a direct integral of irreducible subrepresentations
in which T acts by a constant phase.
2.3 Weyl reflections and integral transforms
Given the principal series representations described in section 2.2, we can ask whether
there exist equivalences between them. Equivalent representations must have the same
eigenvalues of the Casimir operators,15 and these eigenvalues are polynomials in the weights
(∆, ρ) (for SO(d+1, 1)) and (∆, J, λ) (for SO(d, 2)). For example, the quadratic and quartic
Casimir eigenvalues for P∆,J (with trivial λ) are
C2(P∆,J) = ∆(∆− d) + J(J + d− 2),
C4(P∆,J) = (∆− 1)(d−∆− 1)J(2− d− J). (2.28)
The “restricted Weyl group” W ′ is a finite group that acts on these weights, doesn’t mix
discrete and continuous labels, and leaves the Casimir eigenvalues invariant. Conversely,
if two principal series weights have the same Casimirs, they can be related by an element
of W ′.
For example, in the case of SO(d + 1, 1), the restricted Weyl group is W ′ = Z2. Its
non-trivial element SE ∈W ′ acts by
SE(∆, ρ) = (d−∆, ρR), (2.29)
where ρR is the reflection of ρ. Other transformations exist that leave all Casimir eigen-
values invariant, but SE is the only one that does not mix the integral weights of ρ with
the continuous weight ∆.
In the case of SO(d, 2), there are two continuous parameters that can mix, and thus
the restricted Weyl group W ′ is larger. It is isomorphic to a dihedral group of order 8,
15Here we mean all Casimir operators, not just the quadratic Casimir.
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JHEP11(2018)102
w order ∆′ J ′ λ′
1 1 ∆ J λ
S∆ = LSJL 2 d−∆ J λR
SJ 2 ∆ 2− d− J λR
S = (SJL)2 2 d−∆ 2− d− J λ
L 2 1− J 1−∆ λ
F = SJLSJ 2 J + d− 1 ∆− d+ 1 λ
R = SJL 4 1− J ∆− d+ 1 λR
R = LSJ 4 J + d− 1 1−∆ λR
Table 1. The elements of the restricted Weyl group W ′ = D8 of SO(d, 2). Each element w takes
the weights (∆, J, λ) to (∆′, J ′, λ′). The order 2 elements other than S are the four reflection
symmetries of the rectangle, while S is the rotation by π. The center of the group is ZD8 = 1, S.Finally, the element R is a π/2 rotation. The group is generated by L and SJ , with the relations
L2 = S2J = (LSJ)4 = 1.
W ′ = D8.16 This group has a faithful representation on R2 where it acts as symmetries of
the square. Its action on ∆ = d2 + is and J = −d−2
2 + iq can be described by taking s and
q to be Cartesian coordinates in this R2. It is easy to see that this action preserves the
eigenvalues (2.28). Altogether, the elements of W ′ are given in table 1.17
As mentioned above, the representations defined by weights in an orbit of W ′ have
equal Casimir eigenvalues, which means that potentially they can be equivalent. This
indeed turns out to be true [62, 63]. Equivalence of representations means that there exist
intertwining maps between E(∆,ρ) and Ew(∆,ρ), as well as between P(∆,J,λ) and Pw(∆,J,λ) for
all w ∈W ′.The intertwining map between SO(d + 1, 1) representations E∆,ρ and Ed−∆,ρR is well-
known [33, 34, 42]: it is given by the so-called shadow transform
Oa(x) = SE [O]a(x′) ≡∫ddx′〈Oa(x)O†b(x
′)〉Ob(x′). (2.30)
Here O ∈ Ed−∆,ρR , O ∈ E∆,ρ, we use dagger to denote taking the dual reflected repre-
sentation of SO(d), and 〈Oa(x)O†b(x′)〉 is a standard choice of two-point function for the
operators in their respective representations. The integration region is the full Rd (more
precisely, the conformal sphere Sd).
According to our discussion above, in Lorentzian signature there should exist 6 new
integral transforms, corresponding to the other non-trivial elements of W ′. There in fact
exists a general formula for these transforms, valid for any element of W ′ [62, 63].18 How-
16This also turns out to be the Weyl group of BC2 root system, which was recently studied in the context
of conformal blocks in [59, 60]. It would be interesting to better understand the connection of the present
discussion with that work.17To check that the action on λ is as in the table, one can consider the 4d case. The eigenvalues of all
3 Casimirs of SO(2, 4) are written out, for example, in appendix F of [61] with ` = J + λ, ` = J − λ and
λR = −λ. More generally, by solving the system of polynomial equations expressing invariance of these
explicit Casimir eigenvalues, one can check that W ′ is indeed isomorphic to D8.18In the mathematical literature, these transforms are known as Knapp-Stein intertwining operators.
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JHEP11(2018)102
ever, it is most naturally written using a different construction of P∆,J,λ, and the conversion
to the form appropriate for our purposes is cumbersome.19 Thus instead of deriving these
transforms from the general result we will simply give the final expressions and check
that they are indeed conformally-invariant. Furthermore, we will lift these transforms to
representations P∆,J,λ of SO(d, 2).
Although the Lorentzian transforms we define are only necessarily isomorphisms when
acting on principal series representations P∆,J,λ, it is still interesting to consider the analytic
continuation of their action on other representations, like those associated to physical CFT
operators. For example the action of L will be well-defined on physical local operators.
The result of this action will generically be a primary operator with non-integer spin. One
can then ask how such operators make sense in a CFT and what properties do they have.
In this and the following sections we will be able to answer these question by studying the
examples provided by integral transforms. In appendix A we study the same questions on
more general grounds (by using unitarity, positivity of energy, and conformal symmetry)
and reach similar conclusions.
2.3.1 Transforms for S∆, SJ , S
Let us start with the Lorentzian analogue of (2.30). The idea is to essentially keep the
form (2.30) while generalizing to continuous spin,
S∆[O](x, z) ≡ i∫x′≈x
ddx′1
(x− x′)2(d−∆)O(x′, I(x− x′)z), (2.31)
Iµν (x) = δµν − 2xµxνx2
. (2.32)
The integrand is conformally-invariant because I(x− x′) performs a conformally-invariant
translation of a vector at x to a vector at x′. The factor of i is to match a Wick-rotated
version of the Euclidean shadow transform, although we still have SE = (−2)JS∆ after
Wick rotation because of our convention for two-point functions (A.24).
We must specify a conformally-invariant integration region for x′. The essentially
unique choice is to integrate over the region spacelike separated from x. If x is at spatial
infinity ofMd, then this region is the full Poincare patch Md ⊂ Md, and for integer J the
integral is simply the Wick rotation of the Euclidean shadow integral (2.30). If, however,
x is inside the first Poincare patch, then the integral extends beyond the first Poincare
patch on the Lorentzian cylinder Md. All other conformally-invariant regions defined by
x are translations of the spacelike region by powers of T or unions thereof. The two-point
function in these regions differs from the two-point function in the spacelike region only
by a constant phase, and thus the most general choice of S∆ differs from the above by
multiplication by a function of T .20 The possibility of multiplying by a function of Tis present for all the transforms we consider and we just make the simplest choice. The
choice (2.31) is natural because of its relation to (2.30).
19See [42] for an example of this conversion in the case of the shadow transform (2.30).20In particular, there is no ambiguity in representations P∆,J,λ of SO(d, 2).
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JHEP11(2018)102
For SJ , the integral transform is
SJ [O](x, z) ≡∫Dd−2z′(−2 z · z′)2−d−JO(x, z′), (2.33)
where the measure Dd−2z is defined in (2.27). We call this the “spin shadow transform.”
Note that this is essentially the same as the shadow transform in the embedding space [34],
with X replaced by z and d replaced by d− 2.
The transform for S, which we call the “full shadow transform,” is simply the compo-
sition of the commuting transforms for S∆ and SJ ,
S[O](x, z) ≡ (SJS∆)[O](x, z) = i
∫x′≈x
ddx′Dd−2z′(−2 z · z′)2−d−J
(x− x′)2(d−∆)O(x′, I(x− x′)z′)
= (S∆SJ)[O](x, z) = i
∫x′≈x
ddx′Dd−2z′(−2 z · I(x− x′)z′)2−d−J
(x− x′)2(d−∆)O(x′, z′).
(2.34)
These two forms of S are equivalent because I(x− x′)2 = 1, for spacelike x− x′ I(x− x′)is an element of the orthochronous Lorentz group O+(d − 1, 1), and the measure of the
z-integration is invariant under O+(d− 1, 1).
The second line of (2.34) can also be written as
S[O](x, z) = i
∫x′≈x
ddx′Dd−2z′〈OS(x, z)OS(x′, z′)〉O(x′, z′) (2.35)
where OS denotes the representation with dimension d −∆ and spin 2 − d − J . Here, we
are using the following convention for a two-point structure
〈O(x1, z1)O(x2, z2)〉 =(−2z1 · I(x12) · z2)J
x2∆12
, (2.36)
which differs by a factor of (−2)J from some more traditional conventions. Our conventions
for two- and three-point structures are summarized in appendix A.3
2.3.2 Transform for L
The integral transform corresponding to L is
L[O](x, z) =
∫ +∞
−∞dα (−α)−∆−JO
(x− z
α, z). (2.37)
Because it involves integration along a null direction, we call L the “light transform.”
Although most of the transforms in this section are only well-defined on nonphysical repre-
sentations like Lorentzian principal series representations, the light transform is significant
because it can be applied to physical operators as well. Note that it converges near α = ±∞only for ∆ +J > 1.21 In unitary theories it can therefore be applied to all non-scalar oper-
ators and to scalars with dimension ∆ > 1 (which includes all non-trivial scalars in d ≥ 4).
21For Lorentzian principal series Re(∆+J) = 1 but for non-zero Im(∆+J) the integral still makes sense.
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JHEP11(2018)102
x
T x
Figure 4. The contour prescription for the light-transform. The contour starts at x ∈ Md and
moves along the z direction to the point x+ = T x in the next Poincare patch TMd.
Before discussing conformal invariance, let us describe the contour of integration in
more detail. The integral starts at α = −∞, in which case the argument of O is simply x.
It then increases to α = −0, and in the process O moves along z to future null infinity in
Md. As α crosses 0, the integration contour leaves the first Poincare patch Md and enters
the second Poincare patch TMd ⊂ Md. Finally, at α = +∞ it ends at T x ∈ TMd. In
other words, the integration contour is a null geodesic in Md from x to T x with direction
defined by z (figure 4). This is obviously a conformally-invariant contour.
It turns out that no phase prescription is necessary to define (−α)−∆−J for α > 0,
because the naive singularity at α = 0 is cancelled in correlators of O. To see this, note
that (2.37) is equivalent to the following integral in the embedding formalism of [58],
L[O](X,Z) =
∫ +∞
−∞dα (−α)−∆−JO
(X − Z
α,Z
)=
∫ +∞
−∞dαO(Z − αX,−X), (2.38)
where in the second equality we used the homogeneity properties of O(X,Z) in the region
α < 0, together with gauge invariance O(X,Z + βX) = O(X,Z). In (2.38) it is clear that
the point α = 0 is not special (see also appendix B.1 for yet another explanation).
The embedding space integral (2.38) makes conformal invariance of the light-transform
manifest: it is SO(d, 2) invariant, and gauge invariance
L[O](X,Z + βX) = L[O](X,Z) (2.39)
can be proved by shifting α by β in the integral. It is also clear from homogeneity in X and
Z that the dimension and spin of L[O](X,Z) are 1−J and 1−∆, respectively. (Note that
the parameter α carries homogeneity 1 in Z and −1 in X.) Finally, (2.38) confirms the
prescription that the integral goes between x and T x. Indeed, according to the discussion
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JHEP11(2018)102
in section 2.1 the embedding space covers two Poincare patches and T X is simply −X.
The integral in (2.38) starts at the argument Z +∞X which is the same as X modulo R+
and ends at Z −∞X which is −X = T X modulo R+.
Let us describe another way of writing L that will be useful. Equation (2.37) expresses
L in a conformal frame where x is in the interior of a Poincare patch. In this case, the
integration contour extends from one patch into the next. However, if we place x at past
null infinity, the integration contour fits entirely within a single Poincare patch. Specifically,
in the integral (2.38), let us set22
Z = (1, y2, y),
X = (0,−2y · z,−z) (2.40)
to obtain
L[O](x, z) =
∫ ∞−∞
dαO(y + αz, z). (2.41)
Here, x = y −∞z. Equation (2.41) is simply the integral of O along a null ray from past
null infinity to future null infinity, contracted with a tangent vector to the ray. As an
example, the “average null energy” operator is given by
E =
∫ ∞−∞
dαTµν(αz)zµzν = L[T ](−∞z, z), (2.42)
where Tµν is the stress tensor. It follows from our discussion that E transforms like a
primary with dimension −1 and spin 1− d, centered at −∞z.
2.3.3 Transforms for F,R,R
The transforms for the remaining elements F,R,R ∈ D8 are compositions
F ≡ SJLSJ ,
R ≡ SJL,
R ≡ LSJ . (2.43)
For example,
F[O](x,z)≡∫ddζDd−2z′δ(ζ2)θ(ζ0)(−2ζ ·z′)−J−d+2(−2ζ ·z)∆−d+1O(x+ζ,z′) (2.44)
+
∫ddζDd−2z′δ(ζ2)θ(ζ0)(−2ζ ·z′)−J−d+2(−2ζ ·z)∆−d+1(T O)(x−ζ,z′).
22This choice reverses the role of X,Z relative to the usual Poincare section gauge fixing. However, it
still satisfies the required conditions X2 = Z2 = X ·Z = 0. To obtain these expressions, consider the usual
Poicare coordinates for a point shifted by −Lz for large L,
X = (1, (x− Lz)2, x− Lz) ' L× (0,−2x · z,−z),Z = (0, 2z · x, z) = L−1 ×
((1, x2, x)−X
),
from where the new gauge-fixing follows.
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JHEP11(2018)102
Note that here the second term involves an integral over the second Poincare patch
TMd. Similarly to the light transform, here we integrate over all future-directed null
geodesics from x to T x. Because we integrate over all null directions, we call F the
“floodlight transform.”
Similarly, we have
R[O](x, z) =
∫ddζδ(ζ2)θ(ζ0)(−2z · ζ)1−d+∆O(x+ ζ, ζ)
+
∫ddζδ(ζ2)θ(ζ0)(−2z · ζ)1−d+∆(T O)(x− ζ, ζ), (2.45)
R[O](x, z) =
∫dαDd−2z′(−α)−∆−2+d+J(−2 z · z′)2−d−JO
(x− z
α, z′). (2.46)
As an example, R[T ] = SJ [L[T ]] is given by integrating the average null energy oper-
ator E = L[T ] over null directions. This is equivalent to integrating the stress tensor over
a complete null surface, which produces a conformal charge. We can understand this more
formally as follows. Note that the dimension and spin of R[T ] are given by
R(d, 2) = (−1, 1). (2.47)
These are exactly the weights of the adjoint representation of the conformal group. Con-
servation of Tµν ensures that R[T ] transforms irreducibly, so that it transforms precisely
in the adjoint representation. In other words, conservation equation for T becomes the
conformal Killing equation for R[T ]. It can thus be written as a linear combination of
conformal Killing vectors (CKVs):23
R[T ](x, z) = QAwµA(x)zµ
= K · z − 2(x · z)D + (xρzν − xνzρ)Mνρ + 2(x · z)(x · P )− x2(z · P ). (2.48)
Here, A is an index for the adjoint representation of the conformal group, wµA(x) are CKVs,
and the QA are the associated charges. On the second line, we’ve given the charges their
usual names. We can see from (2.48) that inserting R[T ] at spatial infinity x = ∞ gives
the momentum charge. This is a familiar fact from “conformal collider physics” [41].
Similarly, when J is a conserved spin-1 current, R[J ] has dimension-0 and spin-0, which
are the correct quantum numbers for a conserved charge.
2.4 Some properties of the light transform
As noted above, the light transform of the stress-energy tensor is the average null energy
operator L[T ] = E . The average null energy condition (ANEC) states that E is non-
negative,
〈Ψ|E|Ψ〉 ≥ 0. (2.49)
23See [57] for more discussion of writing finite-dimensional representations of the conformal group in
terms of fields on spacetime.
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JHEP11(2018)102
Non-negative operators with vanishing vacuum expectation value 〈Ω|E|Ω〉 = 0 must nec-
essarily annihilate the vacuum |Ω〉 [64].24,25 Indeed, using the Cauchy-Schwarz inequality
for the inner product defined by E , we find
|〈Ψ|E|Ω〉|2 ≤ 〈Ψ|E|Ψ〉〈Ω|E|Ω〉 = 0 (2.50)
for any state |Ψ〉. Thus E|Ω〉 = 0.
In fact, we know that L[O]|Ω〉 = 0 for any local primary operator O — not just the
stress tensor. Indeed, if O has scaling dimension ∆, then L[O] has spin 1 − ∆, which in
a unitary theory is a non-negative integer only if ∆ = 0 or ∆ = 1. However, in these
cases J = 0 and the light transform diverges. For all other scaling dimensions L[O] is a
continuous-spin operator and thus must annihilate the vacuum. This makes it possible for
other null positivity conditions (like those proved in [46] and section 6) to hold as well. In
the rest of this subsection we check explicitly that L[O]|Ω〉 = 0 for all ∆ +J > 1 and make
some general comments about properties of L.
Lemma 2.1. The light transform of a local primary operator, when it exists (i.e. ∆+J > 1),
annihilates the vacuum,26
L[O]|Ω〉 = 0. (2.51)
Proof. We will show that for any local operators Vi,
〈Ω|Vn(xn) · · ·V1(x1)L[O](y, z)|Ω〉 = 0, (2.52)
which implies the result. Let us work in a Poincare patch where y is at past null infinity
and for simplicity assume that the xi fit in this patch; other configurations can be obtained
by analytic continuation. Using a Lorentz transformation we can set z = (1, 1, 0, . . . , 0)
and parameterize the light transform contour as x0 =(v−u
2 , v+u2 , 0, 0, . . .
)for v ∈ (−∞,∞).
We are then computing∫ ∞−∞
dv〈Ω|Vn(xn) · · ·V1(x1)O(x0, z)|Ω〉 =
= limε→+0
∫ ∞−∞
dv〈Ω|Vn(xn − inεe0) · · ·V1(x1 − iεe0)O(x0, z)|Ω〉, (2.53)
where e0 is the future-pointing unit vector in the time direction. The above iε prescription
arranges the operators so that they are time-ordered in Euclidean time, and this is precisely
how the Wightman function should be defined as a distribution. Let us now write
xk − ikεe0 = yk + iζk, k = 0, 1, . . . n, (2.54)
24We thank Clay Cordova for discussion on this point.25Intuitively, the vacuum must contain the same amount of positive-E states and negative-E states in
order for 〈Ω|E|Ω〉 to vanish. Since there are no negative-E states, the vacuum only contains vanishing-Estates and is thus annihilated by E .
26For general spin representations J must be replaced by the sum of all Dynkin labels with spinor labels
taken with weight 12.
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JHEP11(2018)102
ζ0
ζ1
ζ2
ε
Im v > 0
Figure 5. Relationships between the imaginary parts ζk. A deformation of v in the positive
imaginary direction is shown in blue.
where both yk and ζk are real vectors. Positivity of energy implies that Wightman functions
are analytic if ζk is in the absolute future of ζk+1 for all k [65]:27
ζ0 > ζ1 > · · · > ζn. (2.55)
This condition clearly holds when the xk are real. If we then give an arbitrary positive
imaginary part to v while keeping u and other components of x0 fixed, ζ0 = Im(v)z will
remain in the future of ζ1 = −εe0 (see figure 5). Therefore, the integrand is an analytic
function of v in the upper half plane. If we can close the v contour in the upper half plane,
that would imply the required result.
According to the discussion around (2.37), conformal invariance implies that the inte-
gral (2.37) is regular as α→ −0, which in turn implies that the integrand of (2.53) decays
as |v|−∆−J for real v. We will now show that this is also true for complex v in the upper
half-plane, so we can close the contour as long as ∆ + J > 1.
To compute the rate of decay in v, we can use the OPE for the operators Vi, which
converges acting on the left vacuum.28 The leading contribution at large v will be from Oin this OPE, leading to a two-point function of O. Because v is moving in the direction of
its polarization z, the decay of this two-point function is governed not by ∆ but by ∆ + J .
Indeed, we need to consider the two-point function
〈O(0, z′)O(u, v; z)〉. (2.56)
The problem is then essentially two-dimensional: the statement that v is along z means
that O has definite left and right-moving weights of the 2d conformal subgroup. Invariance
27For example, it is easy to check that under this condition (yik + iζik)2 6= 0 for all yik, and thus there
are no obvious null cone singularities. More generally, see appendix A.28For this argument it is important that iε-prescriptions and positive imaginary part of v smear the
operators so that we are working with normalizable states. An argument from the Euclidean OPE is that
the iε shifts separate the operators on the Euclidean cylinder, and Lorentzian times do not affect convergence
of the OPE. The operators in the right hand side of the OPE can be placed anywhere in Euclidean future of
O. Alternatively to (but not logically independently from) the OPE argument, we could have just started
with 〈Ω|OL[O]|Ω〉 in the first place, since states of the form∫ddxf(x)〈Ω|O(x) are dense in the space of
states which can have a non-zero overlap with L[O]|Ω〉.
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JHEP11(2018)102
under the 2d conformal subgroup then selects the component of z′ with the same weights,
so the two-point function is proportional to
〈O(0, z′)O(u, v; z)〉 ∝ (z′1 − z′0)J
u∆−Jv∆+J. (2.57)
Let us see this explicitly in the case of traceless-symmetric tensor O,
〈O(0, z′)O(u, v; z)〉 ∝(z′µI
µν(x0)zν)J
(uv)∆, (2.58)
where we have x0 = 12vz + 1
2uz⊥. Here z⊥ = (−1, 1, 0, . . .) is the basis vector for the u
coordinate and we have (z · z⊥) = 2. The numerator is then
z′µIµν(x0)zν = (z′ · z)− 2u
uv
(1
2(z′ · z)v +
1
2(z′ · z⊥)u
)= (z′1 − z′0)
u
v. (2.59)
This indeed leads to the expected form (2.57).
In summary, we can close the v contour in the upper half plane to give zero whenever
∆ + J > 1.
Recall that the condition ∆ + J > 1 is true for all non-scalar operators in unitary
CFTs, and for all non-identity scalar operators in d ≥ 4 dimensions.
As as simple corollary of lemma 2.1, light transforms of local operators not acting on
the vacuum can be expressed in terms of commutators. For example,
〈Ω|O1L[O3]O2|Ω〉 = 〈Ω|[O1,L[O3]]O2|Ω〉 = 〈Ω|O1[L[O3],O2]|Ω〉. (2.60)
Note that these commutators vanish at spacelike separations, so the integral in the light
transforms only receives contributions from timelike separations. More explicitly, we can
understand the commutators (2.60) as follows. In the integral∫ ∞−∞
dα(−α)−∆−J〈Ω|O1O3(x− z/α, z)O2|Ω〉, (2.61)
there is one singularity in the lower half-plane where 3 becomes lightlike from 1 and another
in the upper half-plane where 3 becomes lightlike from 2 (figure 6). If we deform the
contour to wrap around the first singularity (3 ∼ 1), we obtain the commutator [O1,O3];
if we deform the contour around the second singularity (3 ∼ 2), we obtain [O3,O2].
Lemma 2.1 has the following simple consequence for time-ordered correlators:
Lemma 2.2. Let O be a local primary operator with ∆+J > 1. In a time-ordered correlator
〈V1 . . . VnL[O]〉Ω, (2.62)
if the integration contour of L[O] crosses only past or only future null cones, the transform
is zero. Note that on the Lorentzian cylinder, generically, the contour crosses the null cone
of each Vi exactly once.
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JHEP11(2018)102
α
3 ∼ 1
3 ∼ 2
Figure 6. Contour prescriptions for the α integral in the light transform of a three-point func-
tion (2.60). The black contour corresponds to 〈Ω|O1L[O3]O2|Ω〉, the blue contour corresponds to
〈Ω|[O1,L[O3]]O2|Ω〉, and the red contour corresponds to 〈Ω|O1[L[O3],O2]|Ω〉.
Note that here the notation (2.62) means that L is applied to a physical time-ordered
correlation function, as opposed to time-ordering acting on the continuous spin operator
L[O]. (Since continuous spin operators are necessarily non-local, it is unclear how to
define the latter time-ordering in a Lorentz-invariant way, see appendix A.) We also use
the subscript Ω to stress that we mean a physical correlation function, as opposed to a
conformally-invariant tensor structure.
Finally, let us note that if we use the usual Wightman iε-prescription,29 the light
transform of a Wightman function is an analytic function of its arguments, including the
polarizations. This follows simply from the fact that it is an integral of an analytic function.
This is consistent with our statements concerning analyticity of Wightman functions of
continuous-spin operators in appendix A.
2.5 Light transform of a Wightman function
As a concrete example, and because it will play an important role later, let us compute
the light-transform of the Wightman function
〈0|φ1(x1)O(x3, z)φ2(x2)|0〉 =
(2z · x23 x
213 − 2z · x13 x
223
)Jx∆1+∆2−∆+J
12 x∆1+∆−∆2+J13 x∆2+∆−∆1+J
23
, (2.63)
where φi are scalar operators with dimensions ∆i, and O has dimension ∆ and spin J . (Our
three-point structure normalization differs by a factor of 2J from some more conventional
normalizations. Our conventions are summarized in appendix A.3.) In the above expres-
sion, the Wightman iε prescription is implicit. As discussed at the end of the introduction,
we use the convention that expectation values in the state |Ω〉 denote physical correlation
functions, whereas the expectation values in the state |0〉 denote two- or three-point ten-
sor structures fixed by conformal invariance. The same comment applies to time-ordered
correlation functions 〈· · ·〉Ω and 〈· · ·〉 respectively.
Because the light-transform of a local operator annihilates the vacuum (lemma 2.1), it
is equivalent to the commutators
〈0|φ1L[O]φ2|0〉 = 〈0|φ1[L[O], φ2]|0〉 = 〈0|[φ1,L[O]]φ2|0〉. (2.64)
29In other words, add small Euclidean times to the operators to make the expectation value time-ordered
in Euclidean time.
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JHEP11(2018)102
1 2
3
Figure 7. Causal relationships between points in the light transform (2.64). The original integra-
tion contour is the union of the solid blue line and the dashed line. The solid blue line shows the
region where the commutator [φ1,O] is non-zero.
Specifically, let us compute the third expression above,
〈0|[φ1(x1),L[O](x3,z)
]φ2(x2)|0〉=
∫ +∞
−∞dα(−α)−∆−J〈0|
[φ1(x1),O
(x3−
z
α,z)]φ2(x2)|0〉.
(2.65)
Since the light transform of a Wightman function is analytic (see section 2.4 and
appendix A), we can compute it for any choice of causal relationships, and obtain the answer
for other configurations by analytic continuation. We will work with the configuration in
figure 7. All points lie in a single Poincare patch. The points 1 and 2 are spacelike separated,
and the integration contour starts at 3 < 1 and ends at 3+ > 2. The commutator [φ1,O]
vanishes at spacelike separation, so the upper limit of the integral (2.65) gets restricted to
the value of α when 3 crosses the past null cone of 1.
In our configuration, we have
(z · x13) < 0, (2.66)
−2(z · x13)
x213
< −2(z · x23)
x223
. (2.67)
The first inequality follows because z and x13 are future-pointing and x13 is not null. The
second inequality expresses the fact that the null cone of 1 is crossed before the null cone
of 2.
Taking into account that x213 = eiπ|x2
13| for the ordering φ1O and x213 = e−iπ|x2
13| for
the ordering Oφ1, and restricting the range of integration to the past lightcone of 1, we find
〈0|[φ1(x1),L[O]](x3,z)φ2(x2)|0〉=
=−2isinπ∆1+∆−∆2+J
2
∫ − 2(z·x13)
x213
−∞dα(−α)−∆−J
(2z ·x23x
213−2z ·x13x
223
)Jx∆1+∆2−∆+J
12 |x13′ |∆1+∆−∆2+Jx∆2+∆−∆1+J23′
,
(2.68)
where x′3 = x3 − z/α. Note that the factor (. . .)J in the numerator is independent of α
– 25 –
JHEP11(2018)102
because z is null. We thus need to compute∫ − 2(z·x13)
x213
−∞dα(−α)−∆−J 1
|x13′ |∆1+∆−∆2+Jx∆2+∆−∆1+J23′
=
∫ +∞
2(z·x13)
x213
dα1
|αx213 − 2(z · x13)|
∆1+∆−∆2+J2 (αx2
23 − 2(z · x23))∆2+∆−∆1+J
2
=Γ(∆ + J − 1)Γ
(1− ∆+∆1−∆2+J
2
)Γ(
∆−∆1+∆2+J2
)× 1
|x13|∆1+∆−∆2+Jx∆2+∆−∆1+J23
(2(z · x13)
x213
− 2(z · x23)
x223
)1−∆−J. (2.69)
By (2.66), α has constant sign, which allows us to go to the second line. Because of (2.67),
the function of z which enters (. . .)1−∆−J is positive, so the result is well-defined.
Putting everything together, we find
〈0|φ1(x1) L[O](x3, z)φ2(x2)|0〉
= L(φ1φ2[O])
(2z · x23 x
213 − 2z · x13 x
223
)1−∆
(x212)
∆1+∆2−(1−J)+(1−∆)2 (−x2
13)∆1+(1−J)−∆2+(1−∆)
2 (x223)
∆2+(1−J)−∆1+(1−∆)2
,
(2.70)
where
L(φ1φ2[O]) ≡ −2πiΓ(∆ + J − 1)
Γ(∆+∆1−∆2+J2 )Γ(∆−∆1+∆2+J
2 ). (2.71)
The result (2.70) indeed takes the form of a conformally-invariant correlation function of
φ1 and φ2 with an operator of dimension 1− J and spin 1−∆. Note how continuous spin
structures arise in a natural way from the light transform. Note also that (2.70) is pure
negative-imaginary in the configuration of figure 7, where all quantities in the denominator
are real. This is related to Rindler positivity as we discuss in section 6.1.
Although we did the computation in a specific configuration, we have expressed the
result in terms of an analytic function of the positions. Because the result should be
analytic, the resulting expression (2.70) is valid for any configuration. The iε-prescription
in (2.70) is the same as for the original Wightman function. In particular, if we move x3
back into a configuration where all the points are spacelike separated, we obtain a phase
eiπ∆1+(1−J)−∆2+(1−∆)
2 (2.72)
coming from −x213 becoming negative. This phase will play a role in section 2.7.
2.6 Light transform of a time-ordered correlator
Finally, let us discuss the light-transform of a time-ordered correlator 〈O1O2L[O3]〉. By
lemma (2.2), this is nonzero only if 2− < 3 < 1 (as in figure 7) or 1− < 3 < 2. In
the first nonzero configuration 2− < 3 < 1, the time-ordered correlator is equivalent to
– 26 –
JHEP11(2018)102
the Wightman function 〈0|O1O3O2|0〉 along the entire integration contour of the light
transform. The other nonzero configuration differs by 1↔ 2. Thus, we have
〈O1O2L[O3]〉 = 〈0|O1L[O3]O2|0〉θ(2− < 3 < 1) + 〈0|O2L[O3]O1|0〉θ(1− < 3 < 2). (2.73)
Note that here the standard Wightman functions 〈0|O1O3O2|0〉 and 〈0|O2O3O1|0〉 (on
which the light transforms act) are related to each other by analytic continuation and not
by merely by relabeling the operators in the standard tensor structures 〈0| . . . |0〉.For example, consider the three-point structure (2.63), now assumed to have iε pre-
scriptions appropriate for a time-ordered correlator. From (2.73) and our computation for
the Wightman function (2.70), the light-transform is
〈φ1φ2L[O](x3,z)〉=L(φ1φ2[O]) (2.74)
×[ (
2z ·x23x213−2z ·x13x
223
)1−∆
(x212)
∆1+∆2−(1−J)+(1−∆)2 (−x2
13)∆1+(1−J)−∆2+(1−∆)
2 (x223)
∆2+(1−J)−∆1+(1−∆)2
θ(2−< 3< 1)
+(−1)J
(2z ·x13x
223−2z ·x23x
213
)1−∆
(x212)
∆1+∆2−(1−J)+(1−∆)2 (x2
13)∆1+(1−J)−∆2+(1−∆)
2 (−x223)
∆2+(1−J)−∆1+(1−∆)2
θ(1−< 3< 2).
]The factor of (−1)J in the second term comes from the fact that the original structure
〈φ1φ2O〉 picks up (−1)J when we swap 1↔ 2.30
2.7 Algebra of integral transforms
The L-transformation in (2.70) has the curious property that L2 is a nontrivial function of
∆1,∆2,∆ and J , even though it originates from a Weyl reflection (∆, J)↔ (1− J, 1−∆)
that squares to 1. Specifically, its square acting on a three-point Wightman function is
given by
〈0|φ1(x1) L2[O](x3, z)φ2(x2)|0〉 = α∆1,∆2,∆,J〈0|φ1(x1)O(x3, z)φ2(x2)|0〉, (2.75)
where
α∆1,∆2,∆,J = eiπ∆1+∆−∆2+J
2 L(φ1φ2[OL])× eiπ∆1+(1−J)−∆2+(1−∆)
2 L(φ1φ2[O]) (2.76)
=π
(∆ + J − 1) sinπ(∆ + J)(eiπ(∆1−∆2) − eiπ(∆+J))(eiπ(∆1−∆2) − e−iπ(∆+J)).
The phases in the first line of (2.76) are from (2.72).
Note that the square of the light transform does give back a three-point function of
the same functional form as the original. However, the coefficient α∆1,∆2,∆,J depends on
∆1,∆2 in a non-trivial way that cannot be removed by redefining L by some function of ∆, J
alone. This is in contrast to the Euclidean shadow transform, which squares to a coefficient
N (∆, J) that is independent of the correlation function it acts on (appendix C.2).
30As we explain in appendix A, time-ordered correlators with continuous spin do not make sense, so we
must assume J is an integer in this computation. This means that the factor (−1)J is unambiguous. The
light transform 〈φ1φ2L[O]〉 still gives a sensible continuous-spin structure because the result (2.74) is no
longer a time-ordered correlator, e.g. it has θ-functions.
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JHEP11(2018)102
This “anomaly” in the group relation L2 = 1 occurs for the following reason. The
group-theoretic origin of L only guarantees that it squares to a multiple of the identity when
acting on principal series representations P∆,J defined on the conformal compactification
of Minkowski space Mcd. However, here we are applying it to the space P∆,J defined on
the universal cover Md. The squared transformation L2 still commutes with SO(d, 2), so
it becomes a non-trivial automorphism of the representation P∆,J .
By Schur’s lemma, nontrivial automorphisms can only occur in reducible representa-
tions. Indeed, as discussed in section 2.2, P∆,J is reducible and its irreducible components
are the eigenspaces of T . Within these irreducible components L2 must act by a constant,
and thus we should have
L2 = fL(∆, J, T ). (2.77)
Furthermore, note that L2[O](x, z) only depends on the values of O between x and T 2x.
This means that fL(∆, J, T ) must be at most a quadratic polynomial in T . Finally, because
L2[O] vanishes when acting on the past or future vacuum, fL(∆, J, T ) should have roots at
the eigenvalues of T in O|Ω〉 and 〈Ω|O inside a correlation function,31 which are e±iπ(∆+J).
In fact, as we show explicitly in appendix B.1,
L2 = fL(∆, J, T ) =π
(∆ + J − 1) sinπ(∆ + J)(T − eiπ(∆+J))(T − e−iπ(∆+J)). (2.78)
This immediately implies (2.76) because eiπ(∆1−∆2) is the eigenvalue of t acting on O in
the Wightman function 〈0|φ1(x1)O(x3, z)φ2(x2)|0〉. To see this, write the action of T on
O as
〈0|φ1(x1)T O(x3, z)T −1φ2(x2)|0〉 (2.79)
and use (2.15).
In fact, we can also turn this reasoning around and use the relatively simple computa-
tion (2.76) to fix the polynomial fL(∆, J, T ) in general. This will be helpful in appendix G
where we will need the statement that for general Lorentz irreps ρ the ratio
fL(∆, ρ, T )
(T − γ)(T − γ−1), (2.80)
where γ is the eigenvalue in (2.15) corresponding to (∆, ρ), is independent of T .
More generally, this reasoning implies that relations between restricted Weyl reflections
w ∈ D8 also hold for the corresponding integral transforms, but only up to multiplication
by polynomials in T with coefficients depending on ∆ and J . In the remainder of this
section we derive these modified relations between integral transforms.
First of all, some relations hold by construction given the definitions in section 2.3,
S = SJS∆ = S∆SJ ,
F = SJLSJ ,
R = SJL,
R = LSJ . (2.81)
31Here we need the adjoint action as O → T OT −1, cf. equation (2.15).
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JHEP11(2018)102
Furthermore, we already know that (for simplicity, we consider only P∆,J,λ with trivial λ)
L2 = fL(∆, J, T ), (2.82)
S2J = fJ(J), (2.83)
where we have suppressed the dependence on t. Here fL is a quadratic polynomial in t
defined in (2.78), while fJ(J) depends only on J and is equal to the square of Euclidean
shadow transform in d− 2 dimensions:
fJ(J) =πd−1
(J + d−22 ) sinπ(J + d
2)
1
Γ(−J)Γ(J + d− 2). (2.84)
That is, fJ(J) = N (−J, 0) in d − 2 dimensions, where N (∆, J) in d dimensions is given
in (C.5). These equations allow us to compute
RR = fL(∆, 2− d− J, T )fJ(J), (2.85)
RR = fL(∆, J, T )fJ(1−∆). (2.86)
As we show in appendix B.2, there is another relation,
S∆ = iT −1 LSJL. (2.87)
Together with S = SJS∆ = S∆SJ this implies
S = iT −1 R2 = iT −1 R2, (2.88)
and thus we find
S2 = −T −2R2R2
= −T −2fL(∆, 2− d− J, T )fJ(J)fL(J + d− 1, 1− d+ ∆, T )fJ(1−∆).
(2.89)
Due to S2 = −T −2(SJL)4 = −T −2(LSJ)4, we also have
(LSJ)4 = (SJL)4 = fL(∆, 2− d− J, T )fJ(J)fL(J + d− 1, 1− d+ ∆, T )fJ(1−∆).
(2.90)
At this point it is obvious that fJ and fL completely determine the relations between all
integral transforms, since D8 is generated by L and SJ modulo L2 = S2J = (SJL)4 = 1 and
we have already found the generalization of these relations to the integral transforms L
and SJ in (2.82), (2.83), and (2.90).
A convenient way to summarize these results is by using normalized versions of L
and SJ . Specifically, we define
L ≡ L1
Γ(∆ + J − 1)(T − eiπ(∆+J)), (2.91)
SJ ≡ SJΓ(−J)
πd−2
2 Γ(J + d−22 )
, (2.92)
– 29 –
JHEP11(2018)102
where ∆ and J in there right hand side should be understood as operators reading off
the dimension and spin of the functions they act upon. One can then check the follow-
ing relations
L2 = 1, S2J = 1, (LSJ)4 = (SJ L)4 = 1. (2.93)
These normalized transforms therefore generate the dihedral group D8 without any extra
coefficients. Note that L is very non-local because it has T in the denominator. In par-
ticular, by doing a Taylor expansion in T we see that it involves a sum over an infinite
number of different Poincare patches. Thus, even though L satisfies a simpler algebra, we
mostly prefer to work with L.
3 Light-ray operators
In this section, we explain how to fuse a pair of local operators O1,O2 into a light-ray
operator Oi,J which gives an analytic continuation in spin J of the light-transform of local
operators in the O1 ×O2 OPE. This amounts to defining correlation functions
〈Ω|V1 . . . VkOi,JVk+1 . . . Vn|Ω〉 (3.1)
in terms of those of O1 and O2,
〈Ω|V1 . . . VkO1O2Vk+1 . . . Vn|Ω〉. (3.2)
When J is an integer, Oi,J is related to a local operator in the O1O2 OPE, and these
correlation functions are linked by Euclidean harmonic analysis [42]. Our strategy will be to
start with this relation, rephrase it in Lorentzian signature, and then analytically continue
in J . By the operator-state correspondence, it suffices to consider just two insertions Vi,
and for simplicity we will also restrict to scalars O1 = φ1 and O2 = φ2. (The generalization
to arbitrary spin of O1,O2 will be straightforward.)
3.1 Euclidean partial waves
Consider a Euclidean correlation function 〈φ1φ2V3V4〉Ω, where the V3 and V4 are local
operators of any spin (not necessarily primary) and φ1, φ2 are local primary scalars. By
the Plancherel theorem for SO(d + 1, 1) (due to Harish-Chandra [66]), such a correlation
function can be expanded in partial waves P∆,J that diagonalize the action of the conformal
Casimirs acting simultaneously on points 1 and 2 [42, 67],32,33
〈V3V4φ1φ2〉Ω =
∞∑J=0
∫ d2
+i∞
d2
d∆
2πiµ(∆, J)
∫ddxPµ1···µJ
∆,J (x3, x4, x)〈O†µ1···µJ (x)φ1φ2〉. (3.3)
Here, O has spin J and dimension ∆ ∈ d2 +iR+ on the principal series. The factor µ(∆, J) is
the Plancherel measure (C.5), which we have inserted in order to simplify later expressions.
32For general spin operators we should also include contributions from a discrete series of partial waves.33In [67], the process of forming the Euclidean partial wave P∆,J is called “conglomeration”.
– 30 –
JHEP11(2018)102
For traceless-symmetric O there is no difference between representations O† and O, but
we will keep the daggers in what follows with the view towards the more general case.
Let us make two technical comments about the applicability of this formula. It fol-
lows directly from L2(G) harmonic analysis on SO(d + 1, 1) if ∆1 −∆2 is pure imaginary
(possibly 0) and 〈V3V4φ1φ2〉Ω is square-integrable in the sense that∫ddx1d
dx2 x−2d+4 Re ∆112 〈V3V4φ1φ2〉Ω(〈V3V4φ1φ2〉Ω)∗ <∞. (3.4)
This is precisely the situation when the conformal Casimir operators acting on points 1 and
2 are self-adjoint and we can perform their spectral analysis.34 Neither of these conditions
is satisfied by a typical correlator in a physically-relevant CFT. Lifting the restriction of
square integrability is conceptually easy and is similar to the usual Fourier transform: non-
square integrable correlation functions can be interpreted as distributions (of some kind)
and their partial waves also become distributions.35
Relaxing the restriction ∆1 − ∆2 ∈ iR, on the other hand, seems to be hard to do
from first principles, since the Casimir operators are not self-adjoint anymore. We will
thus not attempt to do this here and instead adopt the following pedestrian approach: we
will imagine multiplying correlation functions by products of scalar two-point functions
xκδijij with κ = 1 so that the scaling dimensions of external operators will formally become
principal series (this will of course modify the conformal block decomposition of these
functions).36 We perform harmonic analysis for these modified functions and then remove
the auxiliary two-point functions by sending κ → 0. For this to make sense we have to
assume that the final expressions can be analytically continued to κ = 0.
With these comments in mind, we may proceed with (3.3). Using the bubble inte-
gral (C.4), we find that P∆,J is given by
Pµ1···µJ∆,J (x3, x4, x) =
(〈φ1φ2O†〉, 〈φ†1φ
†2O〉
)−1
E
∫ddx1d
dx2〈V3V4φ1φ2〉Ω〈φ†1φ†2O
µ1···µJ (x)〉,
(3.5)
where (〈φ1φ2O†〉, 〈φ†1φ
†2O〉
)E
=22J CJ(1)
2dvol(SO(d− 1))(3.6)
34The reason why it is important to have ∆1 − ∆2 ∈ iR is that the adjoint of a Casimir operator acts
on functions with conjugate shadow scaling dimensions ∆∗i . This is a different space of functions than
the one 〈V3V4φ1φ2〉Ω lives in unless ∆∗i = ∆i, which is the case when ∆i ∈ d2
+ iR are principal series
representations. It furthermore turns out that only ∆1 −∆2 is important for the argument, since ∆1 + ∆2
can be changed by multiplying 〈V3V4φ1φ2〉Ω by a two-point function xδ12 for some δ, and such two-point
functions cancel out in equations.35The distributional contribution to the partial wave can be analyzed by subtracting a finite number
of contributions of low dimensional operators to make the function better behaved. This analysis was
essentially performed in [16] and in generic cases amounts to a deformation of ∆-contour in (3.3).36Note that such two-point functions have the right Wightman analyticity properties, and thus do not
spoil the analyticity of physical correlators which we use in the arguments below.
– 31 –
JHEP11(2018)102
is the three-point pairing defined in appendix C.1. In anticipation of performing the light-
transform, let us contract spin indices of O with a null polarization vector zµ to give
P∆,J(x3, x4, x, z) =(〈φ1φ2O†〉, 〈φ†1φ
†2O〉
)−1
E
∫ddx1d
dx2〈V3V4φ1φ2〉Ω〈φ†1φ†2O(x, z)〉, (3.7)
where O(x, z) = Oµ1···µJ (x)zµ1 · · · zµJ .
Physical correlation functions 〈V3V4O∗〉Ω of operators O∗ in the φ1 × φ2 OPE are
residues of the partial waves,
f12∗〈V3V4O∗(x, z)〉Ω = − Res∆=∆∗
µ(∆, J)SE(φ1φ2[O†])P∆,J(x3, x4, x, z)∣∣∣J=J∗
. (3.8)
Here, SE(φ1φ2[O†]) is the shadow transform coefficient (C.7), and f12∗ is the OPE coeffi-
cient of O∗ ∈ φ1 × φ2. Equation (3.8) is a simple generalization of the standard result for
primary four-point functions. We derive it in appendix C.3.
3.2 Wick-rotation to Lorentzian signature
To obtain the promised analytic continuation of L[O], we need to first go to Lorentzian
signature, and then apply the light transform.
We thus Wick-rotate all the operators φ1, φ2, V3, V4,O to Lorentzian signature by set-
ting
τ = (i+ ε)t, (3.9)
where τ and t are Euclidean and Lorentzian time, respectively. In more detail, we simul-
taneously rotate the time coordinates of each of the operators φ1, φ2, V3, V4,O. For the
operators V3, V4,O, this means we analytically continue in the coordinates x3, x4, x. The
operators φ1, φ2 are being integrated over in (3.7), and we rotate their respective inte-
gration contours simultaneously with the analytic continuation of x3, x4, x. Simultaneous
Wick-rotation turns Euclidean correlators into time-ordered Lorentzian correlators. The
result is a double-integral of time-ordered correlators over Minkowski space
P∆,J(x3, x4, x, z) = −(〈φ1φ2O†〉, 〈φ†1φ
†2O〉
)−1
E
∫∞≈1,2
ddx1ddx2〈V3V4φ1φ2〉Ω〈φ†1φ
†2O(x, z)〉.
(3.10)
Here, we have chosen a generic point x∞ on the Lorentzian cylinder Md and written
Minkowski space as the Poincare patch that is spacelike from this point.37,38 All the points
1, 2, 3, 4, x are constrained to lie within this patch. The minus sign in (3.10) comes from
two Wick rotations in the measure dτ1dτ2 = −dt1dt2.
3.3 The light transform and analytic continuation in spin
Let us now move O(x, z) to past null infinity and perform the light transform. We choose
3, 4 such that 3− < x < 4, so that the left-hand side is nonzero, see figure 8a. Since O is
37In particular the result must be independent of which point we choose for x∞. The spurious dependence
of formulas on x∞ will go away soon.38Note that we do not place O(x, z) at infinity before performing the Wick rotation, in contrast to [31].
The reason is that in our case the region of integration for 1, 2 is independent of the position of O so it is
easier to analytically continue in the position of O.
– 32 –
JHEP11(2018)102
∞∞ 4 3
x
x+
1
2
(a) Integration region in (3.12).
∞∞ 4 3
x
x+
1
2
(b) Region S in (3.12) which contributes to the
residue.
Figure 8. The configuration of points within the Poincare patch of ∞. Point 4 is in the future of
x and 3 is in the past of x+, while x is null separated and in the past of ∞. The shaded yellow
(red) region is the region of integration for 1 (2) after taking the light transform, in the first term
in equations (3.11) and (3.12). The dashed null line is spanned by z. Note that in (b), for d > 2
the region S extends in and out of the picture, while the dashed null line doesn’t.
on the Euclidean principal series, the condition Re(∆ + J) > 1 is satisfied and we can plug
in (2.73) to find
L[P∆,J ](x3, x4, x, z) = −(〈φ1φ2O†〉, 〈φ†1φ
†2O〉
)−1
E
×∫
2−<x<1∞≈1,2
ddx1ddx2〈V3V4φ1φ2〉Ω〈0|φ†1L[O](x, z)φ†2|0〉
+ (1↔ 2). (3.11)
See the discussion below (2.73) for the precise meaning of the (1↔ 2) term.
Let us now define
O∆,J(x,z)≡ µ(∆,J)SE(φ1φ2[O†])(〈φ1φ2O†〉,〈φ†1φ
†2O〉
)E
∫2−<x<1∞≈1,2
ddx1ddx2〈0|φ†1L[O](x,z)φ†2|0〉φ1φ2+(1↔ 2).
(3.12)
It is implicit here that x is null separated from ∞. This expression makes sense (at least
formally) for continuous J . The euclidean three-point structure 〈φ†1φ†2O〉 that we started
with is single-valued only for integer J . However, due to the particular Wightman ordering
the structures in (3.12) are well-defined for any J , as discussed in appendix A. In order to
continue to non-integer J , we must also choose an analytic continuation of the prefactors
in (3.12), which we discuss in more detail below. One consequence is that we have two
different analytic continuations: one from even values of J that we denote O+∆,J , and one
from odd values of J that we denote O−∆,J .
– 33 –
JHEP11(2018)102
For integer J , (3.11) and (3.8) imply that the residues O±i,J , defined by
O±∆,J(x, z) ∼ 1
∆−∆±i (J)O±i,J(x, z), (3.13)
have the same three-point functions as light-transforms of local operators in the φ1 × φ2
OPE. (We include a ± subscript on ∆±i (J) because the positions of poles in the (∆, J)
plane are in general different for the even/odd cases.) To be precise, when J is an integer,
the residue of a time-ordered correlator, where time-ordering acts on φ1 and φ2 inside the
definition of O±∆,J ,
〈V3V4O±∆,J(x, z)〉Ω, (3.14)
agrees with
f12O〈V3V4L[Oi,J ]〉Ω, (3.15)
for a local operator Oi,J , where ± is determined by (−1)J = ±1.
We now claim that, for any J , the residue in (3.14) comes from a region S where φ1
and φ2 are simultaneously almost null-separated from x and from each other, see figure 8b.
Indeed, we always expect singularities in correlators when points are null-separated. In
integrated correlators, such singularities can be removed by iε-prescriptions. However,
lightlike singularities in the region S are not removed because they coincide with bound-
aries in the integration regions for x1, x2. In a time-ordered correlator, we can also have
singularities at coincident points. However, we expect singularities related to the φ1 × φ2
OPE to come from 1 being lightlike to 2 and not from other coincident limits.
Let us focus on the first term of (3.12). For this term, it is guaranteed that 1 ≥ 3,
2 ≤ 4, and 1 ≥ 2. In the region S we furthermore have 1 ≤ 4 and 2 ≥ 3, i.e. we have the
ordering 4 ≥ 1 ≥ 2 ≥ 3, and the contribution of the first term of (3.12) to the time-ordered
correlator (3.14) agrees with its contribution to the Wightman function
〈Ω|V4O±∆,JV3|Ω〉. (3.16)
The same obviously holds for the second term, and, moreover, (3.15) agrees with the
Wightman function
f12O〈Ω|V4L[Oi,J ]V3|Ω〉. (3.17)
Since any state in CFT can be approximated by local operators Vi acting on the vacuum in
an arbitrarily small region, this implies that we can interpret (3.12) and (3.13) as operator
equations. Furthermore, by construction, for non-negative integer J we must have, as an
operator equation,
O±i,J = f12OL[Oi,J ] (J ∈ Z≥0, (−1)J = ±1) (3.18)
for some local operator Oi,J .
– 34 –
JHEP11(2018)102
For non-integer J the definition (3.12) with (3.13) provides an analytic continuation
in J of L[Oi,J ]. As we will show in section 4, it is precisely the matrix elements of O±∆,Jand O±i,J which are computed by Caron-Huot’s Lorentzian inversion formula. As discussed
above, the residues O±i,J should only depend on the region of the integral where φ1 and
φ2 are almost null-separated. In fact, it is natural to expect that the residue is further
localized onto the null line defined by z. Hence we refer to them as light-ray operators. In
the next subsection we show this explicitly in the case of mean field theory (MFT).
In our argument for the existence of light-ray operators, it is not necessary that O±∆,J be
a meromorphic function with simple poles. We expect that any non-analyticity in O±∆,J in
the (∆, J) plane should come from the region where φ1 and φ2 are lightlike-separated. Thus,
for example, it should be possible to define light-ray operators by taking discontinuities
across branch cuts of O±∆,J (if they exist). Determining the analyticity structure of O±∆,Jin the (∆, J) plane is an important problem for the future.
As mentioned above, to analytically continue O±∆,J in spin, we must choose an analytic
continuation in J of the prefactors
µ(∆, J)SE(φ1φ2[O†])(〈φ1φ2O†〉, 〈φ†1φ
†2O〉
)E
= (−1)JΓ(J + d
2)Γ(d+ J −∆)Γ(∆− 1)
2πdΓ(J + 1)Γ(∆− d2)Γ(∆ + J − 1)
Γ(∆+J+∆1−∆22 )Γ(∆+J−∆1+∆2
2 )
Γ(d−∆+J+∆1−∆22 )Γ(d−∆+J−∆1+∆2
2 ). (3.19)
Additionally, the term in (3.12) with (1 ↔ 2) has a prefactor differing by (−1)J . Because
of the (−1)J ’s, we must make two separate analytic continuations from even and odd J ,
leading to O±∆,J . In general, we expect the even and odd spectrum of light-ray operators
to be different because they are distinguished by an eigenvalue sO = ±1, as explained in
section 3.3.1. For example, in MFT with a real scalar φ, the analytic continuation of even-J
two-φ operators is nontrivial, but there are no odd-J two-φ operators.
The analytic continuation of the remaining Γ-function factors in (3.19) is determined by
requiring that they be meromorphic and polynomially bounded at infinity in the right half-
plane. This is important for the Sommerfeld-Watson resummation discussed in section 5.2.
The expression (3.19) satisfies these conditions, so provides a good analytic continuation.
When φ1, φ2 are not scalars, then we can relate the prefactor to a rational function of
J times (3.19) using weight-shifting operators [57, 68], and this provides a good analytic
continuation in that case as well.
Although we have assumed scalar φ1, φ2 in this section for simplicity, the generalization
to arbitrary representations O1,O2 is straightforward. We discuss some aspects of the
general case in section 4.2.
3.3.1 More on even vs. odd spin
There is a natural operation that distinguishes even-spin and odd-spin light-ray operators.
First recall that every quantum field theory has an anti-unitary symmetry JΩ = CRT which
– 35 –
JHEP11(2018)102
acts on local operators by [65]
JΩOa(x)J−1Ω =
(iF (e−iπM
01)abOb(x)
)†. (3.20)
Here, x = (−x0,−x1, x2, . . . , xd−1) is the Rindler reflection of x, (Mµν)ab are representation
matrices associated to the Lorentz generators Mµν ,39
[Mµν ,Oa(0)] = −(Mµν)abOb(0), (3.21)
and F is the fermion number of O. We call JΩ “Rindler conjugation” because it is the
modular conjugation operator for the Rindler wedge in the vacuum state [69].40 It is useful
to introduce the notation
O ≡ JΩOJ−1Ω . (3.22)
Note that J2Ω = 1. Furthermore, using (3.22), we clearly have
O1O2 = O1O2, (3.23)
so Rindler conjugation preserves operator ordering.
Rindler conjugation is an anti-unitary symmetry. If we combine it with Hermitian
conjugation, we obtain a linear map of operators
O → O†. (3.24)
This is no longer a symmetry on Hilbert space because it reverses operator ordering. Nev-
ertheless, it makes sense to classify operators into eigenspaces of (3.24). Consider first a
local operator O(x, z) with spin J , and let us set z = z0 = (1, 1, 0, . . . , 0). We have
O(x, z0)†
= O(x, z0) = (−1)JO(x, z0). (3.25)
Integrating x along the z0 direction, we obtain
L[O](−∞z0, z0)†
= (−1)JL[O](−∞z0, z0). (3.26)
For a more general light-ray operator, we have
O(−∞z0, z0)†
= sOO(−∞z0, z0), (3.27)
where now the eigenvalue sO = ±1 is not necessarily related to the quantum number J . If
we obtain O by analytically continuing L[O] from the case where J is even (odd), we will
obtain sO = +1 (−1). In this work, we abuse terminology and refer to light-ray operators
with sO = +1 as “even-spin” and operators with sO = −1 as “odd-spin.”
39The Mµν are antihermitian in our conventions.40The alternative notation CRT comes from the fact that this operator reverses charges and implements a
reflection in both time and a single spatial direction. By contrast the operator CPT implements a reflection
in all spatial directions simultaneously.
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JHEP11(2018)102
3.4 Light-ray operators in Mean Field Theory
In this section we explicitly show that O±i,J are light-ray operators in Mean Field Theory
(MFT). For simplicity, we assume that the scalar operators in (3.12) are distinct fundamen-
tal MFT scalars. More generally, we can imagine that they belong to two decoupled CFTs.
The kernel in (3.12) is obtained from (2.70) by sending x3 to past null infinity according
to the rule
O(−z∞, z) = limL→+∞
L∆+JO(−Lz, z), (3.28)
i.e.
〈0|φ†1L[O]φ†2|0〉 =
= L(φ†1φ†2|O)
2J−1(z · x2 x
21 − z · x1 x
22
)1−∆
(x212)
∆1+∆2+J−∆2 (−z · x1)
∆1−∆2+2−∆−J2 (z · x2)
∆2−∆1+2−∆−J2
. (3.29)
The expression (2.70) was written for 1 > 3, 3 ≈ 2, 1 ≈ 2. With these conditions, the ratio
above is positive. In the integral we need to relax 1 ≈ 2, which is done by adding iε to x02
and −iε to x01, according to the Wightman ordering above. We now introduce lightcone
coordinates by writing
xi =1
2zvi +
1
2z′ui + xi (3.30)
with z′2 = 0, z′ · z = 2 and xi · z = xi · z′ = 0. Since this requires z′ to be past-directed, the
iε-prescription is equivalent to adding a positive imaginary part to u1 and v2 and negative
to u2 and v1. We then find for the integral in the first term of (3.12)
1
4
∫du1du2dv1dv2d
d−2x1dd−2x2
2J−1(u1u2v12+u2x
21−u1x
22
)1−∆φ1(x1)φ2(x2)
(u12v12+x212)
∆1+∆2+J−∆2 (−u1)
∆1−∆2+2−∆−J2 u
∆2−∆1+2−∆−J2
2
.
(3.31)
We have temporarily suppressed the light transform coefficient L(φ†1φ†2[O]).
The integration region has u1 < 0 and u2 > 0. Let us assume for now that v2 > v1
and make the change of variables
u1 = −rα,u2 = r(1− α),
xi = (rv21)12 wi. (3.32)
The integral becomes
1
4
∫ 1
0dα
∫dv1dv2d
d−2w1dd−2w2
2J−1v−1−∆−∆1−∆2+J
221
(α(1− α) + (1− α)w2
1 + αw22
)1−∆
(1 + w212)
∆1+∆2+J−∆2 α
∆1−∆2+2−∆−J2 (1− α)
∆2−∆1+2−∆−J2
×∫ ∞
0
dr
rr−
∆−∆1−∆2−J2 φ1(−rα, v1, (rv21)
12 w1)φ2(r(1− α), v2, (rv21)
12 w2). (3.33)
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JHEP11(2018)102
In the second line, we have isolated the integral∫ ∞0
dr
rr−
∆−∆1−∆2−J2 φ1(−rα, v1, (rv21)
12 w1)φ2(r(1− α), v2, (rv21)
12 w2). (3.34)
The region r ∼ 0 corresponds to φ1 and φ2 being localized near the light ray defined by z.
Now imagine expanding the product of field operators in a power series in r. This
is possible since we have assumed that φ1 and φ2 do not interact and thus there is no
lightcone singularity between them.41 We find terms of the form
rn+m+ 12
(a+b)(−α)n(1− α)mv12
(a+b)
21 wa1wb
2. (3.35)
Only terms with even values of a + b contribute, since the wi integral is invariant under
wi → −wi. Therefore, N = n + m + 12(a + b) ≥ 0 is an integer and the integral over r
takes the form ∫ ∞0
dr
rr−
∆−∆1−∆2−J−2N2 ∼ − 2
∆−∆1 −∆2 − J − 2N. (3.36)
The pole comes from the region of small r. We can see this by imposing an upper cutoff on
r: the residue will be independent of it. (In particular, we can make the cutoff depend on
α and wi thereby cutting out arbitrary regions around the null ray and the residue won’t
change.) The pole is at
∆ = ∆1 + ∆2 + J + 2N, (3.37)
which for integer J are precisely the locations of double-trace operators [φ1φ2]N,J . For
every N , the residue of (3.34) only depends on a finite number of derivatives of φi on the
null ray, and thus is localized on it, as promised in the introduction.
For simplicity, let us focus on the leading twist trajectory with N = 0. The residue
of (3.34) is then
−2φ1(0, v1, 0)φ2(0, v2, 0) (3.38)
and the residue of the integral (3.33) becomes
−1
2
∫ 1
0dα
∫dd−2w1d
d−2w22J−1
(α(1− α) + (1− α)w2
1 + αw22
)1−∆1−∆2−J
(1 + w212)d−∆1−∆2α−∆1+1−J(1− α)−∆2+1−J
×∫dv1dv2(v21 + iε)−1−Jφ1(0, v1, 0)φ2(0, v2, 0). (3.39)
The first line is an overall coefficient which we compute in appendix D and here simply
denote by R(∆1,∆2, J). In the second line, we have restored the iε prescription for vi,
41If we consider φ1 = φ2 = φ, then in MFT we have φ(x1)φ(x2) =:φ(x1)φ(x2) :+〈Ω|φ(x1)φ(x2)|Ω〉. The
singular term is positive-energy in x2 and negative-energy in x1. But in (3.12) we are integrating against
〈0|φ1L[O]φ2|0〉, which has the same energy conditions on x1 and x2. Since the integrals pick out the term
with vanishing total energy in the direction of z for both x1 and x2, the singular piece does not contribute
to (3.12). See also the discussion in section 4.
– 38 –
JHEP11(2018)102
which allows us to relax the assumption v2 > v1. (The factor (v21 + iε)−1−J is understood
to be positive for positive v21 and real J .)
Combining everything together, we conclude that the leading twist operators O0,J are
given by
O0,J(−z∞,z) =i(−1)J
4π
∫dsdt
((t+iε)−1−J+(−1)J(−t+iε)−1−J)φ1(0,s−t,0)φ2(0,s+t,0),
(3.40)
where we have included the contribution of the second term in (3.12), performed the change
of variables v1 = s− t, v2 = s+ t, and used the identity
L(φ†1φ†2[O])R(∆1,∆2, J)
µ(∆, J)SE(φ1φ2[O†])(〈φ1φ2O†〉, 〈φ†1φ
†2O〉
)E
= i(−1)J2J−2
π. (3.41)
The analytic continuations from even and odd J are42
O+0,J(−z∞, z) = +
i
4π
∫dsdt
((t+ iε)−1−J + (−t+ iε)−1−J)φ1(0, s− t, 0)φ2(0, s+ t, 0),
O−0,J(−z∞, z) = − i
4π
∫dsdt
((t+ iε)−1−J − (−t+ iε)−1−J)φ1(0, s− t, 0)φ2(0, s+ t, 0).
(3.42)
These are exactly the null-ray operators advertised in the introduction. We can check that
they are indeed primary by lifting their definitions to the embedding space, where they are
variants of
∼∫ +∞
−∞dαdβ φ1(Z − αX)φ2(Z − βX)(α− β)−J−1. (3.43)
We discuss conformal invariance of this embedding-space integral in the next subsection.
For integer J both kernels for the t-integral are equal to
(t+ iε)−1−J + (−1)J(−t+ iε)−1−J =
=(−1)J
Γ(J + 1)
∂J
∂tJ((t+ iε)−1 − (t− iε)−1
)= −2πi
(−1)J
Γ(J + 1)
∂J
∂tJδ(t). (3.44)
Thus, for integer J we find
O0,J(−z∞, z) =(−1)J
Γ(J + 1)
∫ds
2φ1(0, s, 0)(
↔∂s)
Jφ2(0, s, 0) = L[[φ1φ2]0,J ](−z∞, z). (3.45)
Since total derivatives vanish in the integral over s, it follows that for integer spin O0,J is
given by the light transform of a primary double-twist operator of the form
[φ1φ2]0,J(x, z) ≡ (−1)J
Γ(J + 1)φ1(x)(z · ∂)Jφ2(x) + (z · ∂)(. . .). (3.46)
42It is straightforward to check that O±0,J†
= ±O±0,J .
– 39 –
JHEP11(2018)102
Let us check that these operators are correctly normalized. It was found in [70] that
the full expression for the primary [φ1φ2]0,J is
[φ1φ2]0,J(x, z) = cJ
J∑k=0
(−1)k
k!(J − k)!Γ(∆1 + k)Γ(∆2 + J − k)(z · ∂)kφ1(x)(z · ∂)J−kφ2(x)
(3.47)
and in our case cJ is given by
cJ =(−1)J
Γ(J + 1)
(J∑k=0
1
k!(J − k)!Γ(∆1 + k)Γ(∆2 + J − k)
)−1
. (3.48)
If we write now
〈φ1φ2[φ1φ2]0,J〉Ω = f12J〈φ1φ2OJ〉, (3.49)
and
〈[φ1φ2]0,J [φ1φ2]0,J〉Ω = CJ〈OJOJ〉, (3.50)
where in the right hand side we use the standard structures defined in appendix A.3, then
our normalization conventions are such that CJ/f12J = 1.43 It is a straightforward exercise
to show using (3.47) that
CJf12J
= (−1)JΓ(J + 1)cJ
J∑k=0
1
k!(J − k)!Γ(∆1 + k)Γ(∆2 + J − k)= 1. (3.51)
In doing the calculation it is convenient to use the same null polarization vector for both
operators in (3.50).
3.4.1 Subleading families and multi-twist operators
Although we will not compute the residue of O∆,J for N > 0, let us comment on the form
of the light-ray operators that we expect to obtain, as well as on some further interesting
generalizations. For simplicity, in this section we ignore iε-prescriptions, the difference
between even and odd J , and normalization factors. As mentioned above, the leading
double-twist operators are essentially the primaries
O0,J(X,Z) ≡∫dα dβ φ1(Z − αX)φ2(Z − βX)(α− β)−J−1. (3.52)
The fact that O is a primary follows from conformal invariance of the integral on the right-
hand side. According to the usual rules of the embedding space formalism [58], conformal
invariance is equivalent to
1. homogeneity in X and Z with degrees −∆O and JO, and
2. invariance under Z → Z + λX.
43To be more precise, if O is an operator in φ1 × φ2 OPE, we are computing [φ1φ2]J = f12OO/CO,
which is independent of the normalization of O. Using [φ1φ2]J instead of O then yields the claimed
normalization condition.
– 40 –
JHEP11(2018)102
The former requirement is fulfilled due to homogeneity of the measure dα dβ, the “wave-
function” (α− β)−J−1, and the original primaries φi, which leads to
∆O = 1− J,JO = 1−∆1 −∆2 − J. (3.53)
The latter requirement is satisfied due to translational invariance of the measure dα dβ and
the wavefunction (α− β)−J−1.
This leads to two simple observations. The first is that since the only requirement on
φi is that of being a primary, we can dress them with weight-shifting operators [57]. For
example, let Dm be the Thomas/Todorov differential operator which increases the scaling
dimension of a primary by 1 and carries a vector embedding space index m. Then we
can define
ON,J(X,Z) =
∫dαdβ(Dm1 · · ·DmNφ1)(Z − αX)(Dm1 · · ·DmNφ2)(Z − βX)(α− β)−J−1.
(3.54)
By construction, we now have
∆O = 1− J,JO = 1−∆1 −∆2 − J − 2N. (3.55)
With appropriate iε-prescriptions for α- and β-contours, for integer J these operators
reduce to light transforms of the local family [φ1φ2]N,J . It is clear how (at least in principle)
this construction generalizes to non-scalar φi.
The second observation is that this construction straightforwardly generalizes to multi-
twist operators. In particular, define
Oψ(X,Z) =
∫dα1 · · · dαnφ1(Z − α1X) · · ·φn(Z − αnX)ψ(α1, . . . , αn), (3.56)
where ψ is a wavefunction which is translationally-invariant and homogeneous,
ψ(α1 + β, . . . , αn + β) = ψ(α1, . . . , αn),
ψ(λα1, . . . , λαn) = λ−J−1ψ(α1, . . . , αn). (3.57)
We can easily check that Oψ is a primary with scaling dimension and spin given by
∆O = 1− J,
JO = 1− J +n∑i=1
∆n. (3.58)
Subleading families can be obtained as above, by dressing with weight-shifting operators.
The generalization to non-scalar φi is also clear.
– 41 –
JHEP11(2018)102
∞∞ 4 3
x
x+
1
2
(a) After taking the light transform but before
reducing to a double commutator.
∞∞ 4 3
x
x+
1
2−
2
(b) After reducing to a double commutator.
Figure 9. The configuration of points within the Poincare patch of x∞ at various stages of the
derivation. The blue dashed line shows the support of light transform of O(x, z). The yellow (red)
shaded region shows the allowed region for 1 (2). In the right-hand figure, we indicate that x is
constrained to satisfy 2− < x < 1. Note that after reducing to a double-commutator, the yellow
and red regions are independent of x∞ (as long as x is lightlike from x∞).
4 Lorentzian inversion formulae
In this section we show that matrix elements of O∆,J are computed by a Lorentzian inver-
sion formula of the type discussed by Caron-Huot [16]. Our derivation will borrow some
key steps from [31]. However the light transform will simplify the derivation to the point
where its generalization to external spinning operators is obvious. In particular, after using
the light transform in the appropriate way, it will be immediately clear why the conformal
block GJ+d−1,∆−d+1 and its generalizations appear. For simplicity, we will present most of
the derivation with scalar operators and generalize to spinning operators at the end.
4.1 Inversion for the scalar-scalar OPE
4.1.1 The double commutator
Our starting point is the light-transformed expression (3.11). Let us concentrate on the
first term in (3.11). Because of the restrictions 3− < x < 4 and 2− < x < 1, the lightcone
of x splits Minkowski space into two regions, with 2, 3 in the lower region and 1, 4 in the
upper, see figure 9a. Thus, we can write the integrand as
〈Ω|TV4φ1Tφ2V3|Ω〉〈0|φ†1L[O](x, z)φ†2|0〉. (4.1)
Recall that in our notation, expectation values in the state |Ω〉 denote physical correlation
functions, whereas expectation values in the state |0〉 denote two- or three-point structures
that are fixed by conformal invariance. (For instance, three-point structures 〈0| · · · |0〉 don’t
include OPE coefficients.)
– 42 –
JHEP11(2018)102
We can now use the reasoning in lemma 2.1 to obtain a double commutator.44 Consider
a modified integrand where φ1 acts on the future vacuum,
〈Ω|φ1V4Tφ2V3|Ω〉〈0|φ†1L[O](x, z)φ†2|0〉. (4.2)
Imagine integrating φ1 over a lightlike line in the direction of z, with coordinate v1 along
the line. Because φ1 acts on the future vacuum, the correlator is analytic in the lower half
v1-plane. Furthermore, at large v1, the product of correlators goes like
1
v∆11
× 1
v∆1+∆2+∆+J−2
21
. (4.3)
Here, the first factor comes from the estimate (2.57) of 〈Ω|φ1 · · · |Ω〉 using the OPE and the
second factor comes from direct computation using the three-point function (2.70). Thus,
we can deform the v1 contour in the lower half-plane to give zero whenever
Re(2(d− 2) + ∆1 −∆2 + ∆ + J) > 0. (4.4)
This condition is certainly true for ∆ ∈ d2 + i∞ and J ≥ 0, assuming (for now) that
Re(∆2 −∆1) = 0 (see section 3.1).
Consequently, the x1 integral vanishes if we replace (4.1) with (4.2), so we can
freely replace
TV4φ1 → TV4φ1 − φ1V4 = [V4, φ1]θ(1 < 4). (4.5)
By similar reasoning, we can replace
Tφ2V3 → [φ2, V3]θ(3 < 2). (4.6)
Overall, we find a double commutator in the integrand, together with some extra restric-
tions on the region of integration∫x<1<4
3<2<x+
ddx1ddx2〈Ω|[V4, φ1][φ2, V3]|Ω〉〈0|φ†1L[O](x, z)φ†2|0〉+ (1↔ 2) (4.7)
Note that the spurious dependence on the point at infinity x∞ has disappeared because
the commutators are only nonzero if x < 1 < 4 and 3 < 2 < x+, and these restrictions
imply that 1, 2 lie in the same Poincare patch as 3, 4, x.
In terms of O∆,J we have
〈V4O∆,J(x, z)V3〉Ω =
=µ(∆, J)SE(φ1φ2[O†])(〈φ1φ2O†〉, 〈φ†1φ
†2O〉
)E
∫x<1<4
3<2<x+
ddx1ddx2〈Ω|[V4, φ1][φ2, V3]|Ω〉〈0|φ†1L[O](x, z)φ†2|0〉
+ (1↔ 2). (4.8)
44This argument is the same as the contour manipulation in [31].
– 43 –
JHEP11(2018)102
This gives a Lorentzian inversion formula analogous to the Euclidean inversion for-
mula (3.5). It is different from Caron-Huot’s formula [16] in that it is not formulated
in terms of cross-ratio integrals and it is valid for non-primary or non-scalar Vi. The form
of the inversion formula above will be useful in section 6 where we discuss the average null
energy condition and its generalizations. Note also that the generalization to operators O1
and O2 with nonzero spin is straightforward. In the rest of this subsection we show how
to reduce (4.8) to a cross-ratio integral in the form of [16].
4.1.2 Inversion for a four-point function of primaries
To obtain an integral over cross-ratios, let us specialize to the case where V3 = φ3 and
V4 = φ4 are primary scalars. The partial wave P∆,J in this case is fixed by conformal
invariance up to a coefficient:
µ(∆, J)SE(φ1φ2[O†])P∆,J(x3, x4, x, z) = C(∆, J)〈φ3φ4O(x, z)〉. (4.9)
OPE data is encoded in the resiudes of C(∆, J) by (3.8),
f12O∗f34O∗ = − Res∆=∆∗
C(∆, J∗). (4.10)
The matrix element 〈φ4O∆,J(x, z)φ3〉Ω is the light-transform of (4.9), so (4.8) becomes
C(∆, J)〈0|φ4L[O](x, z)φ3|0〉
= − µ(∆, J)SE(φ1φ2[O†])(〈φ1φ2O†〉, 〈φ†1φ
†2O〉
) ∫x<1<4
3<2<x+
ddx1ddx2〈Ω|[φ4, φ1][φ2, φ3]|Ω〉〈0|φ†1L[O](x, z)φ†2|0〉
+ (1↔ 2). (4.11)
For reasons that will become clear in a moment, let us replace x4 → x+4 (equivalently act
with T4 on both sides). This converts the condition 3− < x < 4 into 3− < x < 4+. At the
same time, let us make the change of variables x2 → x+2 in the integral. We obtain
C(∆, J)〈0|φ4+L[O](x, z)φ3|0〉
= − µ(∆, J)SE(φ1φ2[O])(〈φ1φ2O†〉, 〈φ†1φ
†2O〉
)E
×∫
3−<2<x<1<4+
ddx1ddx2 〈Ω|[φ4+ , φ1][φ2+ , φ3]|Ω〉〈0|φ†1L[O](x, z)φ†
2+ |0〉
+ (1↔ 2). (4.12)
Explicitly, the structure on the left-hand side is (under the additional constraint 3 > 4)
〈0|φ4+L[O](x0, z)φ3|0〉
= L(φ3φ4[O])(−1)J
(2z · x40 x
230 − 2z · x30 x
240
)1−∆
(−x243)
∆4+∆3+J−∆2 (x2
40)∆4−∆3+2−∆−J
2 (x230)
∆3−∆4+2−∆−J2
, (4.13)
– 44 –
JHEP11(2018)102
where L(φ3φ4[O]) is given by (2.71). This expression comes from making the replacements
1, 2, 3→ 3, 4+, 0 in the second line of (2.74) and using x2i4+ = −x2
i4 and z ·x4+0 = −z ·x40.45
Similarly, the structure in the right hand side is
〈0|φ†1 L[O](x0, z) φ†2+ |0〉
= L(φ†1φ†2[O])
(2z · x10 x
220 − 2z · x20 x
210
)1−∆
(−x212)
∆1+∆2+J−∆2 (−x2
10)∆1−∆2+2−∆−J
2 (−x220)
∆2−∆1+2−∆−J2
> 0, (4.14)
which follows from (2.70) by using the same rules.
We would now like to express the coefficient C(∆, J) as an integral of the double-
commutator 〈Ω|[φ4+ , φ1][φ2+ , φ3]|Ω〉 against a conformal block. Both sides of the above
equation transform like conformal three-point functions. We can pick out the coefficient
C(∆, J) by taking a conformally-invariant pairing of both sides with a three-point structure
that is “dual” to the one on the left-hand side.
In other words, in order to isolate C(∆, J), we should find a structure T such that(T, 〈0|φ4+L[O](x, z)φ3|0〉
)L
= 1, (4.15)
with the pairing (·, ·)L defined in equation (E.10) as(〈O3O4O〉, 〈O†3O
†4O
S†〉)L
≡∫
4<3x≈3,4
ddx3ddx4d
dxDd−2z
vol(SO(d, 2))〈O3(x3)O4(x4)O(x, z)〉〈O†3(x3)O†4(x4)OS†(x, z)〉. (4.16)
(Note the causal restrictions in the integral.) It will be convenient to write (4.15) using
the shorthand notation
T = 〈0|φ4+L[O](x, z)φ3|0〉−1. (4.17)
For the pairing (4.15) to be well-defined, 〈0|φ4+L[O]φ3|0〉−1 must transform like a three-
point function with representations 〈φ†4OF†φ†3〉, where OF has dimension and spin
∆OF = J + d− 1,
JOF = ∆− d+ 1. (4.18)
The quantum numbers of OF are precisely those appearing in Caron-Huot’s block. We will
see shortly that this is not a coincidence. Explicitly, the dual structure 〈0|φ4+L[O]φ3|0〉−1
is given by (again for 3 > 4)
〈0|φ4+L[O](x0, z)φ3|0〉−1 (4.19)
=22d−2vol(SO(d− 2))
L(φ3φ4[O])
(−1)J(2z · x40 x
230 − 2z · x30 x
240
)∆−d+1
(−x243)
∆4+∆3−J+∆−2d+22 (x2
40)∆4−J−∆3−∆+2
2 (x230)
∆3−J−∆4−∆+22
.
45These relations follow from the embedding space representation of these quantities as inner products
with X4. An alternative way to obtain this result is to use 〈0|φ4+L[O]φ3|0〉 = 〈0|T φ4T −1L[O]φ3|0〉 =
e−iπ∆4〈0|φ4L[O]φ3|0〉 and then (2.70) with replacements 1 → 4, 2 → 3, 3 → 0, analytically continued.
The factor (−1)J comes from the fact that the standard structure (A.25) depends on a formal ordering of
operators and we need 〈φ3φ4O〉 by convention.
– 45 –
JHEP11(2018)102
4+
4
3
x
1
2
2+
Figure 10. After (temporarily) relabeling the points 2− → 2 and 4− → 4, we have a configuration
where 1 > x > 2 and 3 > 4, with all other pairs of points spacelike separated. This is the same
configuration as in figure 17 of appendix H.2, where we compute the Lorentzian integral for a
conformal block. The integration region for x is shaded yellow. Importantly, it stays away from
3, 4, so the 3→ 4 limit can be computed inside the integrand.
This follows easily from the alternative characterization of the paring (4.16) given in ap-
pendix E.
Finally, pairing both sides of (4.12) with 〈0|φ4+L[O]φ3|0〉−1, we obtain
C(∆, J) =
∫1>23>4
ddx1 · · · ddx4
vol(SO(d, 2))〈Ω|[φ4+ , φ1][φ2+ , φ3]|Ω〉H∆,J(xi) + (1↔ 2), (4.20)
where
H∆,J(xi) = − µ(∆, J)SE(φ1φ2[O])(〈φ1φ2O†〉, 〈φ†1φ
†2O〉
)E
×∫
2<x<1ddxDd−2z〈0|φ†1L[O](x, z)φ†
2+ |0〉〈0|φ4+L[O](x, z)φ3|0〉−1. (4.21)
In the integral for C(∆, J), all the pairs of points xi are spacelike separated except for
1 > 2 and 3 > 4. The causal relations in (4.20) and (4.21) come from the causal relations
in (4.12) and (4.16) which are, together,
4− < 3− < 2 < x < 1 < 4+ < 3+. (4.22)
Recalling that a ≈ b is equivalent to a− < b < a+ (figure 3), we easily find that the above
relations are the same as
1 > x > 2, 3 > 4,
1 ≈ 3, 1 ≈ 4, 2 ≈ 3, 2 ≈ 4. (4.23)
Now the benefit of performing the light-transform becomes clear. The integral (4.21)
over the diamond 2 < x < 1 precisely takes the form of a well-known Lorentzian integral
– 46 –
JHEP11(2018)102
for a conformal block. Note that the integral (4.21) is conformally-invariant and is an
eigenfunction of the conformal Casimir operators acting on points 1, 2 (equivalently 3, 4)
by construction. Importantly, the integral over x stays away from the region near 3, 4, see
figure 10. Thus, we can determine its behavior in the OPE limit 3 → 4 by simply taking
the limit inside the integrand. (This limit corresponds to the Regge limit of the physical
operators at 1, 2+, 3, 4+.) Any eigenfunction of the conformal Casimirs is fixed by its OPE
limit, so this determines the full function. Thus, it’s clear that H∆,J is proportional to a
conformal block, with external operators φ†1, . . . , φ†4, and an exchanged operator with the
quantum numbers of OF†.
We perform this analysis in detail in appendix H.2. Using the result (H.37), we find
H∆,J(xi) =q∆,J
(−x212)
∆1+∆22 (−x2
34)∆3+∆4
2
(x2
14
x224
) ∆2−∆12
(x2
14
x213
) ∆3−∆42
G∆iJ+d−1,∆−d+1(χ, χ),
(4.24)
where
q∆,J = −(−1)J22d−2vol(SO(d− 2))(〈φ1φ2O†〉, 〈φ†1φ
†2O〉
)E
µ(∆, J)SE(φ1φ2[O])L(φ1φ2[O])
L(φ3φ4[O])b∆1,∆2
J+d−1,∆−d+1
= −22dvol(SO(d− 2))Γ(∆+J+∆1−∆2
2 )Γ(∆+J−∆1+∆22 )Γ(∆+J+∆3−∆4
2 )Γ(∆+J−∆3+∆42 )
16π2Γ(∆ + J)Γ(∆ + J − 1).
(4.25)
(The quantity b∆1,∆2
∆,J is defined in (H.36) and the conformal block G is defined in ap-
pendix H.1.) Factors other than b∆1,∆2
∆,J come from (4.21) and the structures (4.14)
and (4.19). In the proof of the Lorentzian inversion formula in [31], performed without
using the light transform, one obtains an expression for H∆,J as an integral over a region
totally spacelike from 1, 2+, 3, 4+, which is harder to understand.
4.1.3 Writing in terms of cross-ratios
Finally, let us replace 2+ → 2 and 4+ → 4 so that the physical operators are again at the
points 1, 2, 3, 4. The inversion formula reads
C(∆, J) =
∫4>12>3
ddx1 · · · ddx4
vol(SO(d, 2))〈Ω|[φ4, φ1][φ2, φ3]|Ω〉(T −1
2 T−1
4 H∆,J(xi)) + (1↔ 2). (4.26)
Here, T −1i denotes a shift xi → x−i or, more generally, application of the T −1 to the
operator at i-th position. In the integrand, we can isolate quantities that depend only on
cross-ratios, times a universal dimensionful factor |x12|−2d|x34|−2d,
〈Ω|[φ4, φ1][φ2, φ3]|Ω〉(T −12 T
−14 H∆,J(xi)) =
1
|x12|2d|x34|2d(4.27)
× 〈Ω|[φ4, φ1][φ2, φ3]|Ω〉T∆i(xi)
G∆iJ+d−1,∆−d+1(χ, χ),
– 47 –
JHEP11(2018)102
where
T∆i(xi) ≡1
|x12|∆1+∆2 |x34|∆3+∆4
(|x14||x24|
)∆2−∆1(|x14||x13|
)∆3−∆4
. (4.28)
Since we now have a fixed causal ordering of the points, we do not have to worry about
an iε prescription in these expressions and we can simply take absolute values of space-
time intervals.
We can gauge-fix (4.26) to obtain an integral over cross-ratios alone. As explained
in [31],46 the measure becomes∫ddx1 · · · ddx4
vol(SO(d, 2))
1
|x12|2d|x34|2d→ 1
22dvol(SO(d− 2))
∫ 1
0
∫ 1
0
dχdχ
χ2χ2
∣∣∣∣χ− χχχ
∣∣∣∣d−2
. (4.29)
Putting everything together, we find
C(∆,J) =q∆,J
22dvol(SO(d−2))
[∫ 1
0
∫ 1
0
dχdχ
χ2χ2
∣∣∣χ−χχχ
∣∣∣d−2 〈Ω|[φ4,φ1][φ2,φ3]|Ω〉T∆i(xi)
G∆iJ+d−1,∆−d+1(χ,χ)
+(−1)J∫ 0
−∞
∫ 0
−∞
dχdχ
χ2χ2
∣∣∣χ−χχχ
∣∣∣d−2 〈Ω|[φ4,φ2][φ1,φ3]|Ω〉T∆i(xi)
G∆iJ+d−1,∆−d+1(χ,χ)
].
(4.30)
Here, G∆,J(χ, χ) denotes the solution to the Casimir equation that behaves as
(−χ)∆−J
2 (−χ)∆+J
2 for negative cross-ratios satisfying |χ| |χ| 1. This precisely co-
incides with Caron-Huot’s Lorentzian inversion formula.
4.1.4 A natural formula for the Lorentzian block
To make it easy to generalize the above result to arbitrary representations, let us write it in
a more transparent way. First we need to introduce more flexible notation for a conformal
block. Let
〈O1O2O〉〈O3O4O〉〈OO〉
(4.31)
denote the conformal block formed by gluing the three-point structures in the numerator
using the two-point structure in the denominator. We describe the gluing procedure in more
detail in appendix H.1. In particular, the gluing procedure is well-defined (for a restricted
causal configuration) even if O is a continuous-spin operator. Using this notation, the
coefficient function C(∆, J) is defined by
〈φ1φ2φ3φ4〉Ω =
∞∑J=0
∫ d2
+i∞
d2−i∞
d∆
2πiC(∆, J)
〈φ1φ2O〉〈φ3φ4O〉〈OO〉
, (4.32)
where O has dimension ∆ and spin J .
46We use a definition of the measure on SO(d, 2) which differs from the one [31] by a factor of 2d.
– 48 –
JHEP11(2018)102
Using the same notation, we claim that the function H∆,J(xi) in (4.26) is given by
H∆,J(xi) = − 1
2πi
(T2〈φ1φ2L[O]〉)−1(T4〈φ3φ4L[O]〉)−1
〈L[O]L[O]〉−1, (1 > 2, 3 > 4). (4.33)
In the numerator, (T2〈φ1φ2L[O]〉)−1 is the dual structure to T2〈φ1φ2L[O]〉 via the three-
point pairing (E.10). It is given by (4.19), with the replacement 3, 4 → 1, 2. Note that
while we have written the structures in the numerators in terms of light transforms of time
ordered products, they can alternatively be written in terms of Wightman functions for
the kinematics we are considering, since
T2〈φ1φ2L[O]〉 = T2〈0|φ2L[O]φ1|0〉 (when 1 > 2, 1, 2 ≈ 0),
T4〈φ3φ4L[O]〉 = T4〈0|φ4L[O]φ3|0〉 (when 3 > 4, 3, 4 ≈ 0). (4.34)
The structure 〈L[O]L[O]〉−1 in the denominator is dual to the double light-transform of
the time-ordered two-point function 〈OO〉 via the conformally-invariant two-point pairing,(〈L[O]L[O]〉−1, 〈L[O]L[O]〉
)L
= 1. (4.35)
Here the pairing (·, ·)L for two-point functions is defined in (E.3). In order for the pair-
ing in (4.35) to be conformally-invariant, 〈L[O]L[O]〉−1 must transform like a two-point
function of OF.
We have already computed the three-point structures in the numerator, so to ver-
ify (4.33), we need to compute 〈L[O]L[O]〉. Here, it is important to treat two-point
structures as distributions. By lemma 2.2, 〈O(x1, z1)L[O](x2, z2)〉 vanishes if x2 > x1
or x2 < x1 — i.e. it vanishes almost everywhere. However, it is nonzero if x1 is precisely
lightlike from x2. Specifically, 〈O(x1, z1)L[O](x2, z2)〉 is a distribution localized where x2
is on the past lightcone of x1.47 In fact, it is proportional to the integral kernel for the
“floodlight transform” F.
Let us now actually compute 〈L[O]L[O]〉. It is useful to think of this structure as an
integral kernel K, defined by
(Kf)(x, z) ≡∫ddx′Dd−2z′ 〈L[O](x, z)L[O](x′, z′)〉f(x′, z′). (4.36)
In (4.36), we can integrate one of the L-transforms by parts, giving
(Kf)(x, z) =
∫ddx′Dd−2z′ 〈L[O](x, z)O(x′, z′)〉(T −1L[f ])(x′, z′). (4.37)
To simplify (4.37) further, we can express the time-ordered two-point function 〈OO〉in terms of integral transforms and use the algebra derived in section 2.7. When x, x′ are
spacelike, 〈O(x, z)O(x′, z′)〉 is precisely the kernel for S. However, S is supported only in
the region x ≈ x′, whereas the time-ordered two-point function has support everywhere.
47Note that this is different from treating two-point functions as physical Wightman functions, so there
is no contradiction with previous discussion.
– 49 –
JHEP11(2018)102
More precisely, keeping track of the phases as we move x, x′ into different Poincare patches,
we have
〈O(x, z)O(x′, z′)〉 =(−2z · z′(x− x′)2 + 4z · (x− x′)z′ · (x− x′))J
((x− x′)2 + iε)∆+J
= S
(1 +
∞∑n=1
e−inπ(∆+J)T n +∞∑n=1
e−inπ(∆+J)T −n)
= S−2iT sinπ(∆ + J)
(T − eiπ(∆+J))(T − e−iπ(∆+J)). (4.38)
Plugging this into (4.37), we find
K = LS−2iT sinπ(∆ + J)
(T − eiπ(∆+J))(T − e−iπ(∆+J))T −1L
= S−2i sinπ(∆ + J)
(T − eiπ(∆+J))(T − e−iπ(∆+J))L2
=−2πi
∆ + J − 1S, (4.39)
where in the second line we used that L,S, T commute with each other, together with
the formula L2 = fL(J + d − 1,∆ − d + 1, T ), where fL is given in equation (2.78). The
arguments of fL come from the fact that K acts on a representation with dimension J+d−1
and spin ∆− d+ 1.
The kernel of S in the last line is the two-point function of an operator with spin 1−∆
and dimension 1− J . Thus, using our two-point pairing (E.3), we find
〈L[O]L[O]〉−1 = −∆ + J − 1
2πi22d−2vol(SO(d− 2))〈OFOF〉, (4.40)
where 〈OFOF〉 is the standard two-point structure (A.24) for an operator with dimension
J + d − 1 and spin ∆ − d + 1. Combining this with the three-point structures in the
numerator, and comparing with the result (4.24) for H∆,J(xi), we verify (4.33).
Note that (4.33) is independent of a choice of normalization of the integral transform
L. In fact, it depends only on the three-point structures 〈φ1φ2O〉, 〈φ3φ4O〉, the two-point
structure 〈OO〉, and the existence of a conformally-invariant map between representations
P∆,J,λ and P1−J,1−∆,λ (which L implements). The formula would still be true if we chose
different normalization conventions for two and three-point functions, because this would
change the definition of C(∆, J) in a compatible way, via (4.32). Because it is essentially
independent of conventions, we call (4.33) a “natural” formula.
4.2 Generalization to arbitrary representations
4.2.1 The light transform of a partial wave
The derivation in the previous section is straightforward to generalize to the case of arbi-
trary conformal representations φi → Oi. In this case, three-point functions admit multiple
– 50 –
JHEP11(2018)102
conformally-invariant structures 〈O1O2O〉(a), so partial waves PO,(a) carry an additional
structure label.48 They are defined by
〈V3V4O1O2〉Ω =∑ρ,a
∫ d2
+i∞
d2
d∆
2πiµ(∆, J)
∫ddxPO,(a)(x3, x4, x)〈O†(x)O1O2〉(a). (4.41)
(Here, we implicitly contract the SO(d) indices of PO,(a) and the operator O†.)The logic leading to the double-commutator integral (4.7) is essentially unchanged.
We find
L[PO,(a)](x3, x4, x, z)
= −(〈O1O2O†〉(a), 〈O†1O†2O〉
(b))−1E
×∫x<1<4
3<2<x+
ddx1ddx2〈Ω|[V4,O1][O2, V3]|Ω〉〈0|O†1L[O](x, z)O†2|0〉
(b)
+ (1↔ 2), (4.42)
where (〈O1O2O†〉(a), 〈O†1O†2O〉(b))
−1E is the inverse of the three-point pairing (C.2) de-
fined by
(〈O1O2O†〉(a), 〈O†1O†2O〉
(b))−1E (〈O1O2O†〉(c), 〈O†1O
†2O〉
(b))E = δca (4.43)
4.2.2 The generalized Lorentzian inversion formula
To generalize the remaining steps leading to the Lorentzian inversion formula, we seemingly
need to understand of all the factors entering the expression for H∆,J(xi) (4.24). However,
this is unnecessary because the generalization is obvious from the natural formula (4.33).
The coefficient function Cab(∆, ρ) we would like to compute is defined by
〈O1 · · · O4〉Ω =∑ρ,a,b
∫ d2
+i∞
d2−i∞
d∆
2πiCab(∆, ρ)
〈O1O2O†〉(a)〈O3O4O〉(b)
〈OO†〉, (4.44)
where O has dimension ∆ and SO(d)-representation ρ. Here, we sum over principal series
representations E∆,ρ, as well as three-point structures a, b. The obvious generalization
of (4.20) and (4.33) is
Cab(∆, ρ) = − 1
2πi
∫4>12>3
ddx1 · · · ddx4
vol(SO(d, 2))〈Ω|[O4,O1][O2,O3]|Ω〉
× T −12 T
−14
(T2〈O1O2L[O†]〉(a)
)−1 (T4〈O4O3L[O]〉(b))−1
〈L[O]L[O†]〉−1
+ (1↔ 2). (4.45)
The dual structures in the numerator are defined by((T2〈O1O2L[O†]〉(a)
)−1, T2〈O1O2L[O†]〉(c)
)L
= δca,((T4〈O4O3L[O]〉(b)
)−1, T4〈O4O3L[O]〉(d)
)L
= δdb , (4.46)
48The possible structures in a three-point function of spinning operators are classified in [71].
– 51 –
JHEP11(2018)102
where (·, ·)L is the three-point pairing defined in (E.10). The two-point structure in the
denominator is the dual of 〈L[O]L[O†]〉 via the two-point pairing (E.3).
Note that the structure(T2〈O1O2L[O†]〉(a)
)−1transforms like a three-point function
of representations 〈O†1O†2O†F〉 and similarly for the operators 3 and 4. In (4.45), we are
implicitly contracting Lorentz indices of Oi with their dual indices in these structures.
4.2.3 Proof using weight-shifting operators
Equation (4.45) follows if we prove the generalization of the expression (4.33) for H, with
H defined using the appropriate generalization of (4.21). Specifically, the definition of
H becomes
H∆,ρ,(ab)(xi) = −µ(∆, ρ†)SE(O1O2[O†])ca(〈O1O2O†〉(c), 〈O†1O†2O〉
(d))−1 (4.47)
×∫
2<x<1ddxDd−2z〈0|O†1L[O](x, z)O†
2+ |0〉(d)(〈0|O4+L[O](x, z)O3|0〉(b))−1.
We want to prove that
H∆,ρ,(ab)(xi) = − 1
2πi
(T2〈O1O2L[O†]〉(a)
)−1 (T4〈O4O3L[O]〉(b))−1
〈L[O]L[O†]〉−1. (4.48)
Our proof will proceed in two steps. Here we are going to show that if for a given ρ (4.48)
is valid for some “seed” choice of SO(d) irreps of external operators, it is then valid for
all choices of external irreps. In appendix G using methods of [57] we show that validity
of (4.48) for traceless-symmetric ρ implies its validity for seed blocks for all ρ. Together
these statements imply (4.48) in full generality.
Generalizing the external representations. It is convenient to consider the structure
defined by
Ta ≡ µ(∆, ρ†)SE(O1O2[O†])ca(〈O1O2O†〉(c), 〈O†1O†2O〉
(d))−1E 〈O
†1O†2+O〉(d). (4.49)
We can check that
Ta = (〈O†O〉, 〈O†O〉)E(〈O1O2SE [O†]〉(a))−1E , (4.50)
where all pairings and inverses are Euclidean. Indeed, we can compute the Euclidean
paring
(Td, 〈O1O2SE [O†]〉(a))E = SE(O1O2[O†])ab(Ta, 〈O1O2O†〉(b))E= µ(∆, ρ†)SE(O1O2[O†])abSE(O1O2[O†])bd= µ(∆, ρ†)N (∆, ρ†)δad = (〈O†O〉, 〈O†O〉)Eδ
ad . (4.51)
Here we used the relation (C.8) between the Plancherel measure and the square of the
Euclidean shadow transform. Importance of the structures Ta comes from the fact that it
is the light transform of their Wick rotation which enters (4.47).
– 52 –
JHEP11(2018)102
We now choose some other SO(d) irreps ρ′1 and ρ′2 for operators O′1 and O′2 such that
there is a unique tensor structure49
〈O′1O′2O†〉. (4.52)
We then can write
Ta = (〈O†O〉, 〈O†O〉)ET −12 D12,aT2(〈O′1O′2SE [O†]〉)−1
E , (4.53)
where D12,a are contractions of weight-shifting operators acting on points 1 and 2 [57, 72].50
We can use this to write
H∆,ρ,(ab)(xi) = D12,aH′∆,ρ,(b)(xi), (4.54)
where H ′ is given by (4.47) with O′1 and O′2 instead of O1 and O2, and using the unique
tensor structure on the left of H ′.
On the other hand, we can write
δad =1
(〈O†O〉, 〈O†O〉)E(Td, 〈O1O2SE [O†]〉(a))E
= (T −12 D12,dT2(〈O′1O′2SE [O†]〉)−1, 〈O1O2SE [O†]〉(a))E
= ((〈O′1O′2SE [O†]〉)−1, (T −12 D12,dT2)∗〈O1O2SE [O†]〉(a))E , (4.55)
where we integrated the differential operators T −12 D12,dT2 by parts inside the Euclidean
pairing. This produces new operators D∗12,d, which are again contractions of weight-shifting
operators.51 We thus conclude that
(T −12 D12,dT2)∗〈O1O2SE [O†]〉(a) = δad〈O′1O′2SE [O†]〉. (4.56)
Canceling SE on both sides (it is invertible on generic tensor structures) we find
(T −12 D12,dT2)∗〈O1O2O†〉(a) = δad〈O′1O′2O†〉. (4.57)
We now want to show that
D12,a(T2〈O′1O′2L[O†]〉)−1L = (T2〈O1O2L[O†]〉(a))−1
L (4.58)
where the inverse structure is understood with respect to Lorentzian pairing. This follows
by doing the above calculation in reverse and in Lorentzian signature. First, we apply L
to both sides of (4.57) and use T ∗ = T −1,
T −12 D
∗12,dT2〈O1O2L[O†]〉(a) = δad〈O′1O′2L[O†]〉. (4.59)
49In odd dimensions and for fermionic ρ the number of tensor structures is always even, and so it is not
possible to make this choice. However, there we can make a choice such that there is only one parity-even
structure, which will be good enough.50Note that T −1
2 D12,dT2 are differential operators which can be interpreted in Euclidean signature. In
particular, if D12,d = D1,ADA2 for A transforming in an irreducible representation W of the conformal group
then T −12 D12,dT2 is proportional to D12,d with coefficient equal to the eigenvalue of T in W .
51For details see appendix F and [57, 68].
– 53 –
JHEP11(2018)102
Then, we apply T2 to both sides and take Lorentzian contraction with (T2〈O′1O′2L[O†]〉)−1L
((T2〈O′1O′2L[O†]〉)−1L ,D∗12,dT2〈O1O2L[O†]〉(a))L = δad , (4.60)
and finally integrate by parts,
(D12,d(T2〈O′1O′2L[O†]〉)−1L , T2〈O1O2L[O†]〉(a))L = δad . (4.61)
This is equivalent to (4.58) The crucial point here is that integration by parts leads to the
same operation on the weight-shifting operators both in Euclidean and Lorentzian signature
(on integer-spin operators). A way to summarize this calculation is by saying that
(T2〈O1O2L[O†]〉)−1L and T2(〈O1O2SE [O†]〉)−1
E (4.62)
have the same transformation properties under weight-shifting operators acting on 1 and 2.
This implies that if (4.48) is true for O′1 and O′2, it is also true for O1 and O2, since we
can simply apply D12,a in both (4.47) and (4.48). Since exactly the same tensor structure
appears for the operators O3,O4 in (4.47) and (4.48), an analogous (even simpler) argument
works for this tensor structure as well. In conclusion, if (4.48) holds for a seed conformal
block, it holds for all conformal blocks with the same ρ.
5 Conformal Regge theory
5.1 Review: Regge kinematics
Consider a time-ordered four-point function of scalar operators 〈φ1 · · ·φ4〉. Its conformal
block expansion in the 12→ 34 channel takes the form
〈φ1(x1) · · ·φ4(x4)〉=∑∆,J
p∆,JG∆i∆,J(xi) (5.1)
=1
(x212)
∆1+∆22 (x2
34)∆3+∆4
2
(x2
14
x224
)∆2−∆12
(x2
14
x213
)∆3−∆42 ∑
∆,J
p∆,JG∆i∆,J(χ,χ),
where p∆,J are products of OPE coefficients. This expansion is convergent whenever χ, χ ∈C\[1,∞) [73]. However, it fails to converge in the Regge limit.52
To reach the Regge regime, which was originally described for CFT correlators in [3],
let us place the operators in a 2d Lorentzian plane with lightcone coordinates
x1 = (−ρ,−ρ),
x2 = (ρ, ρ),
x3 = (1, 1),
x4 = (−1,−1). (5.2)
52The other OPE channels 14 → 23 and 13 → 24 are still convergent, though they are approaching the
boundaries of their regimes of validity, as discussed in the introduction.
– 54 –
JHEP11(2018)102
1
2
4 3
ρ
ρ
Figure 11. The Regge limit in the configuration (5.2). We boost points 1 and 2 while keeping
points 3 and 4 fixed. This configuration is related by an overall boost to the one in figure 1.
The usual cross-ratios are given by
χ =4ρ
(1 + ρ)2, χ =
4ρ
(1 + ρ)2. (5.3)
It is also useful to introduce polar coordinates
ρ = reiθ = rw, ρ = re−iθ = rw−1. (5.4)
In Euclidean signature, r and θ are real. By contrast in Lorentzian signature, r is real,
θ becomes pure-imaginary (it is conjugate to a boost), and ρ, ρ become independent real
variables. To reach the Regge regime, we apply a large boost to operators 1 and 2, while
keeping 3 and 4 fixed (figure 11). More precisely, we take
θ = it+ ε, (t→∞), (5.5)
so that
ρ = re−t+iε, ρ = ret−iε, (t→∞). (5.6)
Here, we use the correct iε prescription to compute a time-ordered Lorentzian correlator
when t > 0. With this prescription, the cross-ratios behave as follows. As t-increases, χ
moves toward zero. Meanwhile, χ initially increases, then goes counterclockwise around 1,
and finally decreases back to zero (figure 12).
The only difference between the Regge and 1→ 2 OPE limits from the perspective of
the cross-ratios χ, χ is the continuation of χ around 1. In both cases, we take χ, χ → 0.
This is because the Regge limit resembles an OPE limit between points in different Poincare
patches. This observation was made in [37]. Specifically, the configuration in figure 11 is
related by a boost to the one in figure 13. The Regge limit can thus be described as 1→ 2−
and 3 → 4−. The cross-ratios χ, χ are unchanged when we apply T to any of the points,
which is why they still go to zero in this limit.
– 55 –
JHEP11(2018)102
χ, χ
0 1χ
χ
Figure 12. The paths of the cross ratios χ, χ when moving from the Euclidean regime to the Regge
regime. In the Euclidean regime, χ, χ are complex conjugates (gray points). As we boost x1, x2, the
cross ratio χ decreases towards zero, while χ moves counterclockwise around 1 before decreasing
towards zero. For sufficiently large t, χ follows the same path as χ, but we have separated the paths
to clarify the figure.
1
24
3
2− 4−
Figure 13. Another description of the Regge limit is x1 → x−2 and x3 → x−4 . The points x−2 , x−4
are shown in gray. The cross-ratios χ, χ associated with the points 1, 2, 3, 4 are the same as those
associated with 1, 2−, 3, 4−.
To understand what happens to the conformal block expansion (5.1) in the Regge
regime, we must compute the monodromy of G∆i∆,J(χ, χ) from taking χ counterclockwise
around 1. This was described in [16]. Firstly, we have the decomposition
G∆i∆,J(χ, χ) = gpure
∆,J (χ, χ) +Γ(J + d− 2)Γ(−J − d−2
2 )
Γ(J + d−22 )Γ(−J)
gpure∆,2−d−J(χ, χ), (5.7)
where gpure∆,J is the solution to the conformal Casimir equation defined by
gpure∆,J (χ, χ) = χ
∆−J2 χ
∆+J2 × (1 + integer powers of χ/χ, χ) (χ χ 1). (5.8)
For small χ, gpure∆,J has a simple form in terms of a hypergeometric function [74],
gpure∆,J (χ, χ) = χ
∆−J2 k∆+J(χ)× (1 +O(χ)) (χ 1), (5.9)
k2h(χ) = χh2F1
(h− ∆12
2, h+
∆34
2, 2h, χ
), (5.10)
where ∆ij ≡ ∆i−∆j . The monodromy of gpure∆,J as χ goes around 1 can then be determined
from (5.9) using elementary hypergeometric function identities, keeping χ small so that the
approximation (5.9) remains valid.
– 56 –
JHEP11(2018)102
Let us defer discussing the precise form of the monodromy until section 5.3, and focus
on one important feature. Note that k2h(χ) is a conformal block for SL(2,R). In particular,
it is a solution to the conformal Casimir equation (a second-order differential equation) with
eigenvalue h(h − 1). Under monodromy, it will mix with the other solution, which differs
by h→ 1− h. In terms of ∆, J , this becomes
(∆, J)→ (1− J, 1−∆), (5.11)
i.e. it is the affine Weyl reflection associated to the light transform. After monodromy, in
the limit χ, χ→ 0 each block contains a term
χ∆−J
2 χ1−∆+1−J
2 ∼ e(J−1)t (t 1). (5.12)
In other words, the monodromy of each block grows as e(J−1)t in the Regge limit. Because
the sum (5.1) includes arbitrarily large J , the OPE expansion formally diverges as t→∞.
In what follows, it will be important to understand the large-J limit of conformal
blocks in slightly more detail. We compute this in appendix H.3:
gpure∆,J (χ, χ) ∼
4∆f1−∆(12(r + 1
r ))w−J
(1− w2)d−2
2 (r2 + 1r2 − w2 − 1
w2 )12
((1− r
w )(1− rw)
(1 + rw )(1 + rw)
)∆12−∆342
(|J | 1),
(5.13)
where w = eiθ and f1−∆(x) is given in (H.43). For us, the most important feature of (5.13)
is that its J-dependence is w−J . Note that the small-w limit of (5.13) is consistent with
the claim that gpure∆,J grows as w1−J = e(J−1)t in the limit t→∞.
5.2 Review: Sommerfeld-Watson resummation
Taking the monodromy of χ around 1 requires leaving the region |ρ| < 1 where the sum
over ∆ in the conformal block expansion converges. The conformal partial wave expansion
gives a way to avoid this problem: we replace a sum of the form∑
∆ |ρρ|∆/2 with an
integral over ∆ ∈ d2 + iR. This integral is better-behaved when |ρ| > 1.
In the Regge limit we still have the problem that each individual block grows like
e(J−1)t. This can be dealt with in a similar way: by replacing the sum over J with an
integral in the imaginary direction. This trick is called the Sommerfeld-Watson transform.
Let us begin with the conformal partial wave expansion
〈φ1(x1) · · ·φ4(x4)〉 =∞∑J=0
∫ d2
+i∞
d2−i∞
d∆
2πiC(∆, J)F∆i
∆,J(xi),
F∆i∆,J(xi) ≡
1
2
(G∆i
∆,J(xi) +SE(φ1φ2[O])
SE(φ3φ4[O])G∆id−∆,J(xi)
). (5.14)
For integer J , the coefficient function C(∆, J) can be written
C(∆, J) = Ct(∆, J) + (−1)JCu(∆, J), (J ∈ Z), (5.15)
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JHEP11(2018)102
where Ct comes from the first term in the Lorentzian inversion formula (4.30), and Cu
comes from the second term with 1 ↔ 2. (The superscripts t and u stand for “t-channel”
and “u-channel.”) Each of the functions Ct,u(∆, J) has a natural analytic continuation in
J that is bounded in the right half-plane. This follows from (4.30), since the conformal
block G∆iJ+d−1,∆−d+1(χ, χ) is well-behaved in the square χ, χ ∈ [0, 1] when J is in the right
half-plane.
Let us split the partial wave F∆i∆,J into two pieces
F∆i∆,J(xi) = F∆,J(xi) +H∆,J(xi), (5.16)
where F∆,J behaves like w−J at large J ,
F∆,J(xi) ≡1
(x212)
∆1+∆22 (x2
34)∆3+∆4
2
(x2
14
x224
)∆2−∆12
(x2
14
x213
)∆3−∆42
× 1
2
(gpure
∆,J (χ, χ) +SE(φ1φ2[O])
SE(φ3φ4[O])gpured−∆,J(χ, χ)
), (5.17)
and H∆,J(xi) represents the remaining terms, which behave like wJ+d−2 at large J . We
must treat the two terms in (5.16) differently in the Sommerfeld-Watson transform. Let
us focus on the first term. The sum over integer spins can be written as a contour integral
∞∑J=0
C(∆, J)F∆,J(xi) = −∮
ΓdJCt(∆, J) + e−iπJCu(∆, J)
1− e−2πiJF∆,J(xi)
(Re(θ) ∈ (0, π), Im(θ) = 0), (5.18)
where the contour Γ encircles all the nonnegative integers clockwise. Here, we have carefully
chosen the analytic continuation of C(∆, J) so that the integrand is bounded at large J in
the right half-plane whenever θ satisfies the given conditions. For this, we use the fact that
F∆,J(xi) behaves as w−J at large J . Because the other term in (5.16) behaves as wJ+d−2
at large J , we must replace e−iπJ → eiπJ to get an integral for that term that is valid in
the same range of θ.
The contour integral (5.18) is more suitable than a naıve sum over spins for continuing
to the Regge regime. Recall that the issue with a sum over J was that a conformal block
with spin J grows as e(J−1)t in the Regge limit. Because the integrand in (5.18) is well-
behaved at large J , we can deform the contour Γ to a region where Re(J) < 1, so that its
contributions die as t → ∞.53 In doing so, we may pick up new poles in Cu,t(∆, J) with
real part Re(J) > 1. The rightmost such pole will dominate the correlator in the Regge
limit. Denote the deformed contour, including these new poles, by Γ′ (figure 14).
After deforming the contour, we now have a representation of the correlator that is
valid in the strip
Re(θ) ∈ (0, π), Im(θ) > 0, (5.19)
53A natural choice is the Lorentzian principal series Re(J) = − d−22
.
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JHEP11(2018)102
J
Γ
Γ′
j(ν)
Figure 14. Integration contours in the J plane. The contour Γ (blue) encircles all the integers
clockwise. The deformed contour Γ′ runs parallel to the imaginary axis, asymptotically approaching
Re(J) = −d−22 at large imaginary J . In deforming the contour, we must ensure that Γ′ avoids non-
analyticities, like a pole at non-integer J , branch cuts, or other singularities. Here, we show a single
non-integer pole at J = j(ν) and possible non-analyticities in the shaded region. However, this is
only an example — we don’t know the structure of the J-plane in general.
which includes the angle θ = it + ε required for a time-ordered Lorentzian correlator.
Thus, we can continue to the Regge regime. The continuation of H∆,J(xi) does not give a
growing contribution in the Regge limit, so let us ignore it for the moment. We find that
the four-point function behaves as
〈φ1(x1) · · ·φ4(x4)〉 ∼ −∮
Γ′dJ
∫ d2
+i∞
d2−i∞
d∆
2πi
Ct(∆, J) + e−iπJCu(∆, J)
1− e−2πiJF∆,J(xi)
, (5.20)
where F∆,J(xi) denotes the continuation to Regge kinematics, including the monodromy
of χ around 1 and phases arising from the prefactor in (5.17).54
In planar large-N theories, the rightmost feature of Γ′ is conjectured to be an isolated
pole J = j(ν) where ∆ = d2 + iν. Assuming this is the case, we obtain
〈φ1(x1) · · ·φ4(x4)〉 ∼ −2πi
∫ ∞−∞
dν
2πResJ=j(ν)
Ct(d2 + iν, J) + e−iπJCu(d2 + iν, J)
1− e−2πiJFd
2 +iν,J(xi)
.
(5.21)
5.3 Relation to light-ray operators
The appearance of the affine Weyl transform (5.11) is suggestive that Regge kinematics
should be related to the light transform and light-ray operators. To see how, let us finally
54Representing the correlator as an integral over both ∆ and J is natural from the point of view of
Lorentzian harmonic analysis, where principal series representations are labeled by continuous ∆ = d2
+ is
and J = − d−22
+ it. However, it is not immediately obvious how the representation (5.20) is related to the
Plancherel theorem for SO(d, 2). We leave this question for future work.
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JHEP11(2018)102
compute F∆,J(xi) using (5.9). We find
F∆,J(xi) = − iπΓ(∆ + J)Γ(∆ + J − 1)
Γ(∆+J+∆122 )Γ(∆+J−∆12
2 )Γ(∆+J+∆342 )Γ(∆+J−∆34
2 )T∆i(xi)G1−J,1−∆(χ, χ)
+ . . . , (5.22)
where T∆i(xi) is the product of |xij |’s given in (4.28). Here, we have explicitly written
the term that is growing in the Regge limit. The “. . . ”’s represent other solutions of
the Casimir equations that do not grow in the Regge limit, coming from both F∆,J and
H∆,J . The above expression is valid in the configuration 4 > 1, 2 > 3, with other points
spacelike-separated.
Comparing with (4.24) and (4.33), we immediately recognize
F∆,J(xi) =
1
2T −1
2 T−1
4
(T2〈φ1φ2L[O†]〉)(T4〈φ3φ4L[O]〉)〈L[O]L[O†]〉
+ . . . , (5.23)
where we use the notation for a conformal block introduced in section 4.1.4. Equation (5.23)
is the main observation of this section. In the case where Regge kinematics is dominated
by an isolated pole (5.21), the residue ResJ=j(ν) means that coefficients in the integrand
can be interpreted as products of OPE coefficients for light-ray operators. This is be-
cause a nontrivial residue comes from the neighborhood of the light ray.55 Plugging (5.23)
into (5.21), we find a sum/integral of conformal blocks for these light-ray operators.
In the gauge-theory literature, the object that controls the Regge limit of a planar
amplitude is called the “Pomeron” [75, 76]. Here, we see that for planar CFT correlation
functions, the Pomeron is a light-ray operator: it is proportional to the rightmost residue
in J of O∆,J , for ∆ ∈ d2 + iR.
The observation (5.23) also lets us immediately generalize conformal Regge theory to
arbitrary operator representations. In the Regge limit, we have
〈O1(x1) · · · O4(x4)〉 ∼ −1
2
∑λ,a,b
∮Γ′dJ
∫ d2
+i∞
d2−i∞
d∆
2πi
Cab(∆, J, λ)
1− e−2πiJ(5.24)
× T −12 T
−14
(T2〈O1O2L[O†]〉(a))(T4〈O3O4L[O]〉(b))〈L[O]L[O†]〉
.
Here, Cab(∆, J, λ) is the unique analytic continuation of Cab(∆, ρ) such that Cab(∆,J,λ)1−e−2πiJ e
−iθJ
is bounded for large J in the right-half plane and θ ∈ (0, π). The weight J is the length of
the first row of the Young diagram of ρ, and λ represents the remaining weights of ρ, as
discussed in section 2.2. The indices a, b run over three-point structures.
As before, it is straightforward to argue that (5.24) is the only possibility consistent
with the scalar case and with weight-shifting operators. It would be interesting to verify
it more directly, and in general to characterize all monodromies of blocks in terms of the
integral transforms in section 2.3. Note that (5.24) displays a beautiful duality with the
generalized Lorentzian inversion formula (4.45).
55The same is true if the Regge limit is dominated by a cut instead of a pole, though now we have a
doubly-continuous family of light-ray operators, parameterized by ν and J along the cut.
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JHEP11(2018)102
We can try to interpret (5.23) as a contribution to the non-vacuum OPE of φ1φ2 in
the following way. We construct light-ray operators as an integral of the form (1.15), which
together with conformal symmetry implies that we should be able to write, schematically,
φ1φ2 =
∫dν Bν,j(ν)[O0,j(ν)] + other contributions. (5.25)
Here B is a kind of OPE kernel which is fixed by conformal symmetry, and the equation
should be interpreted in an operator sense. The representation (5.23) suggests that (5.25)
is a good version of the OPE in non-vacuum states, with the first term giving the only
possibly-growing contribution in the Regge limit.
The “other contributions” can perhaps be understood by studying the terms that we
ignored above, coming form H∆,J and part of F∆,J . We expect that they can be understood
more systematically using harmonic analysis on the Lorentzian conformal group SO(d, 2).
(We hope to address this in future work.) In a finite-N CFT, the correlator saturates
in the Regge limit — i.e. it eventually stops growing. Thus, the details of these terms
will presumably be important for determining the actual behavior of the correlator in the
Regge limit.56
6 Positivity and the ANEC
The average null energy condition (ANEC) states that E = L[T ] is a positive-semidefinite
operator. The ANEC was proven in [45] using information theory and in [46] using causal-
ity. The causality-based proof [46] proceeds by isolating the contribution of E in a cor-
relation function and using Rindler positivity to show that the contribution is positive.
Isolating E requires using the OPE outside its naıve regime of validity. However, the au-
thors of [46] give an argument that one can still trust the leading term in the OPE in an
asymptotic expansion in the lightcone limit.
From our work in section 3, we now have an alternative construction of E as a special
case of a light-ray operator. Using this construction, we can avoid asymptotic expansions
and any technical issues associated with using the OPE outside its regime of validity.
Beyond technical convenience, our approach gives extra flexibility. The authors of [46] also
prove a higher-spin version of the ANEC:
EJ ≡ L[XJ ] ≥ 0, (J = 2, 4, . . . ), (6.1)
where XJ is the lowest-dimension operator with spin J .57,58,59 Our construction lets us
generalize this statement to
EJ ≥ 0, (J ∈ R≥Jmin), (6.2)
56We thank Sasha Zhiboedov for discussions on this point.57More precisely, XJ can be the lowest-dimension operator with spin J in any OPE of the form O† ×O.58The higher-spin version of the ANEC was first discussed in [77], where it was also proven for sufficiently
high spin.59The proof of the higher-spin ANEC in [46] relies on some assumptions about subleading terms when
the OPE is used as an asymptotic expansion outside of its regime of convergence. We thank Tom Hartman
for discussion on this point.
– 61 –
JHEP11(2018)102
where Jmin ≤ 1 is the smallest value of J for which the Lorentzian inversion formula
holds [16]. Here, EJ(x, z) denotes the light-ray operator with dimension and spin (1 −J, 1−∆), where ∆, J are real and ∆ is minimal. This result follows by writing a sum rule
for all light-ray operators, and simply observing that it is positive by Rindler positivity
when (∆, J) satisfy the above conditions. When J is an integer, (6.2) reduces to (6.1).
However, when J is not an integer, (6.2) is a new condition.
A possible connection between Lorentzian inversion formulae and the ANEC was first
suggested by Caron-Huot using a toy dispersion relation [16]. In this section, we are simply
making the connection more precise.
6.1 Rindler positivity
Rindler positivity is a key ingredient in the causality-based proof of the ANEC [46], so let
us review it. Given x = (t, y, ~x) ∈ Rd+1,1, define the Rindler reflection
x = (t, y, ~x) = (−t∗,−y∗, ~x). (6.3)
Rindler conjugation defined in (3.20) and (3.22) maps an operator O in the right Rindler
wedge to an operator O in the left Rindler wedge. For traceless-symmetric tensors, it is
given by
O(x, z) = O†(x, z). (6.4)
The statement of Rindler positivity is that
〈Ω|O1 · · · OnO1 · · · On|Ω〉 ≥ 0, (6.5)
where Oi are restricted to the right Rindler wedge
WR = (u, v, ~x) : uv > 0, arg v ∈ (−π2 ,
π2 ), ~x ∈ Rd−2. (6.6)
(Here, we use lightcone coordinates u = y − t, v = y + t.)
To establish (6.5) for general causal configurations of the Oi, [78] appeals to Tomita-
Takesaki theory. However, this is not necessary as argued in [46]. We can summarize their
argument as follows. Because the operators O1 · · · On act on the vacuum, we can perform
the OPE to replace
O1 · · · On|Ω〉 =∑OC(xi, x, ∂x)O(x)|Ω〉, (6.7)
where C(xi, x, ∂x) is a differential operator. We are free to choose x to be any point in
WR (we cannot choose x to be timelike from the xi). Truncating the sum, we approximate
the right hand side by a local operator. The expectation value (6.5) then becomes a
Rindler-reflection symmetric two-point function. Positivity of this two-point function is a
consequence of reflection-positivity, since the two points are spacelike-separated.
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JHEP11(2018)102
6.2 The continuous-spin ANEC
Following [46], we will prove
i〈Ω|V E ′JV |Ω〉 ≥ 0, (6.8)
where V is any local operator located at a point xV = (0, δ,0) ∈ WR in the right Rindler
wedge. Here, E ′J is a continuous-spin light-ray operator of spin-J with lowest twist, oriented
along the null direction z = (1, 1,~0). As argued in [46], it follows that E ′J satisfies the
positivity condition
eiπ2J〈Ω|(R · V )†(t = −iδ) E ′J (R · V )(t = iδ)|Ω〉 ≥ 0, (6.9)
where R rotates by π2 in the Euclidean yτ -plane, with τ = it, and R · V represents the
action of R on V at the origin. States of the form (R · V )(t = iδ)|Ω〉 ∈ H are dense in H,
by the state-operator correspondence. Thus,
EJ ≡ eiπ2JE ′J (6.10)
is a positive operator.
Let φ be a real scalar primary. We will produce E ′J by smearing two φ insertions. For
simplicity, we will not attempt to divide by OPE coefficients in the φ × φ OPE. Thus,
when J is an integer, we will actually have EJ = fφφXJL[XJ ], where XJ is the lowest-twist
operator of spin-J in the φ× φ OPE and fφφXJ is an OPE coefficient. In particular E2 in
this section differs from the usual ANEC operator by a factor of fφφT .
From (4.8), we have
i〈VO+∆,J(−∞z, z)V 〉 =
∫−∞z<x1<xVxV <x2<∞z
ddx1ddx2〈Ω|[V , φ(x1)][φ(x2), V ]|Ω〉K∆,J(x1, x2),
K∆,J(x1, x2) =2iµ(∆, J)SE(φφ[O])
(〈φφO〉, 〈φφO〉)E〈0|φ(x1)L[O](−∞z, z)φ(x2)|0〉. (6.11)
We have included a factor of 2 from the term 1 ↔ 2 in (4.8), and we should interpret the
prefactors in K∆,J as being analytically continued from even J . The matrix elements of
EJ are defined by
i〈Ω|V E ′JV |Ω〉 = Res∆=∆∗
i〈VO+∆,J(−∞z, z)V 〉, (6.12)
where ∆∗ is the location of the pole in O+∆,J with minimal real ∆. The expression (6.11)
is guaranteed to be convergent for ∆ ∈ d2 + iR on the principal series. In particular it
converges at ∆ = d2 . Our strategy will be to show that i〈VO+
∆,J(x, z)V 〉 is strictly negative
as we move rightward along the real axis starting from ∆ = d2 (figure 15). It follows that
the first pole we encounter must have positive residue.60
60Requiring negativity for all ∆ between d2
and the first pole is stronger than necessary. It should be
possible to improve our proof by establishing negativity only for ∆ sufficiently close to the first pole.
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JHEP11(2018)102
∆
i〈VO+∆,JV 〉
∆ = d2
negative
positive residue
Figure 15. We show that i〈VO+∆,JV 〉 is negative for ∆ between d
2 (the principal series) and the
first pole. It follows that the first pole has positive residue.
The kernel K∆,J is given by
K∆,J(x1, x2) =2J iµ(∆, J)SE(φφ[O])L(φφ[O])
(〈φφO〉, 〈φφO〉)E(z · x2x
21 − z · x1x
22)1−∆
x2∆φ−∆+J12 (−z · x1)
2−J−∆2 (z · x2)
2−J−∆2
.
(6.13)
We would like to show that K∆,J(x1, x2) is a positive-definite kernel when integrated against
Rindler-symmetric configurations of x1, x2. Note that this is a stronger condition than
K∆,J(x, x) ≥ 0 point-wise.
Consider first an inversion x 7→ x′ = xx2 that places EJ at null infinity. In this conformal
frame, the three-point structure 〈0|φL[O]φ|0〉 becomes translationally invariant. Thus our
kernel should be a translationally-invariant function of x′1, x′2, times some scale-factors that
depend independently on x1, x2. Indeed, it is easy to check(z · x2 x
21 − z · x1 x
22
)1−∆
x2∆φ−∆+J12 (−z · x1)
2−J−∆2 (z · x2)
2−J−∆2
= x′2∆φ
1 x′2∆φ
2
(−z · x′1
)J+∆−22
(z · x′2
)J+∆−22
(z · (x′2 − x′1))1−∆
(x′2 − x′1)2∆φ−∆+J. (6.14)
Because our kernel originates from the light-transform of a three-point structure, it in-
herits Rindler positivity properties. These are made clear by going to a kind of complexified
Fourier-space in the inverted coordinates x′i. Define lightcone coordinates x− = u = y − tand x+ = v = y+t. One can prove the following identity which is valid in the right Rindler
wedge u, v > 0:
u1−∆
(uv + ~x2)2∆φ−∆+J
2
=22−2∆φ−J
πd−2
2 Γ(2∆φ−∆+J
2 )Γ(2∆φ+J+∆−d
2 )
×∫k>0
ddk (−k2)2∆φ+∆+J−d−2
2 (−k−)1−∆fk(x)
fk(x) ≡ e−12k+u+ 1
2k−v+i~k·~x. (6.15)
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JHEP11(2018)102
Here, the notation k > 0 indicates that k is restricted to the interior of the forward null cone.
This ensures that k+u is positive and k−v is negative, so that the integral is convergent.
The complexified plane wave fk(x) is designed to satisfy
fk(x)∗ = fk(−x). (6.16)
Putting everything together, we find
K∆,J(x1, x2) = K∆,J
∫k>0
ddk (−k2)2∆φ+∆+J−d−2
2 (−k−)1−∆ψk(x2)(ψk(x1))∗, (6.17)
where
ψk(x) ≡ 1
x2∆φ
( ux2
)J+∆−22
exp
(−1
2k+u+ 1
2k−v + i~k · ~x
x2
), (6.18)
K∆,J =21−d−∆+J+2∆φΓ(J + d
2)Γ(J+d+1−∆2 )Γ(∆− 1)
π3(d−1)
2 Γ(J + 1)Γ(d+J−∆2 )Γ(∆− d
2)Γ(J+∆+d−2∆φ
2 )Γ(J−∆+2d−2∆φ
2 ). (6.19)
Consequently, we can write
i〈VO+∆,J(−∞z, z)V 〉 = −K∆,J
∫k>0
ddk (−k2)2∆φ+∆+J−d−2
2 (−k−)1−∆〈ΘkΘk〉,
Θk =
∫xV <x<∞z
ddxψk(x)[φ(x), V ]. (6.20)
The coefficient K∆,J is positive whenever
∆− J < d, and ∆− J < 2(d−∆φ). (6.21)
This is also the condition for K∆,J(x1, x2) to be integrable without an iε prescription. When
these conditions hold, the minus sign in (6.20) ensures that the first nontrivial residue in
∆ is positive. This proves the ANEC and its continuous spin generalization in this case.
Let us understand the condition ∆−J < 2(d−∆φ) in more detail. When this inequality
fails, two things happen. Firstly, the factor
Γ
(J −∆ + 2d− 2∆φ
2
)(6.22)
in K∆,J may no longer be positive. Secondly, the kernel K∆,J(x1, x2) develops a naively
non-integrable singularity along the lightcone. To make sense of this singularity, one must
take into account the appropriate iε prescription for x1, x2. This turns K∆,J(x1, x2) into
a non-sign-definite distribution, and then we cannot conclude anything about the sign
of (6.20). To get the strongest result, we should pick φ to be the lowest-dimension scalar
in the theory. The spin-2 ANEC then follows if ∆φ ≤ d+22 . Large-spin perturbation
theory [17–26, 79] and Nachtmann’s theorem [11, 18, 80, 81] imply that the minimum twist
∆− J at each spin J is always less than 2∆φ. Thus, we can ensure ∆ − J < 2(d−∆φ) if
∆φ ≤ d2 . This condition is also sufficient to ensure ∆ − J < d. Thus, the continuous-spin
ANEC follows if ∆φ ≤ d2 .
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JHEP11(2018)102
6.3 Example: Mean Field Theory
The continuous spin version of ANEC is easy to check in MFT. (This is essentially the same
calculation as in [46, 82].) We have already computed the leading twist operators E ′J = O+0,J
in section 3.4. In this section we need the straightforward generalization of (3.42) to the
case of identical operators,
E ′J = O+0,J =
i
2π
∫dsdt(t+ iε)−1−J :φ
(s+ t
2z
)φ
(s− t
2z
):, (6.23)
with a future-directed null z. We can explicitly compute these operators in terms of
creation-annihilation operators using
φ(x) = N− 1
2∆φ
∫p>0
ddp
(2π)d|p|∆φ− d2
(a†(p)e−ipx + a(p)eipx
), (6.24)
where ∆φ is the scaling dimension of φ and
N∆ =22∆−1π
d−22
(2π)dΓ(∆)Γ(∆− d−2
2 ) > 0. (6.25)
The creation-annihilation operators satisfy the commutation relation
[a(p), a†(q)] = (2π)dδd(p− q). (6.26)
Plugging (6.24) into (6.23), we find
E ′J =iN− 1
2∆φ
2π
∫p,q>0
ddp
(2π)dddq
(2π)d
∫dsdt(t+ iε)−1−J
[a†(p)a†(q)e−
i2 (p+q)·zs− i2 (p−q)·zt
+ a(p)a(q)ei2 (p+q)·zs+ i
2 (p−q)·zt
+ a†(p)a(q)e−i2 (p−q)·zs− i2 (p+q)·zt
+a†(q)a(p)ei2 (p−q)·zs+ i
2 (p+q)·zt]. (6.27)
The first two terms under the integral vanish because s-integration restricts (p+ q) · z = 0,
which is impossible since both p and q are in the forward null cone. This is consistent with
the requirement that O+0,J should annihilate both past and future vacua. Since (p+q)·z < 0
we can close the t-contour in the upper half-plane for the third term (for J > 0) and thus
it also vanishes. We are left with the last term, where we can close the t-contour in the
lower half-plane. Specifically, we get for s and t integrals∫dsdt(t+ iε)−1−Je
i2 (p−q)·zs+ i
2 (p+q)·zt =2π2δ((p− q) · z)e−
iπ2
(J+1)
Γ(J + 1)
(−(p+ q) · z
2
)J.
(6.28)
Combining with the rest of the expression we find, using the lightcone coordinates p =
zpv/2− z′pu/2 + p with z · z′ = 2,
E ′J =πe−
iπ2JN− 1
2∆φ
Γ(J + 1)
∫ ∞0
dpupJuA†(pu)A(pu), (6.29)
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JHEP11(2018)102
where
A(pu) ≡∫|p|<pupv
dpvdd−2p
(2π)da(pu, pv,p). (6.30)
For EJ = eiπ2JE ′J we then obtain
EJ =πN− 1
2∆φ
Γ(J + 1)
∫ ∞0
dpupJuA†(pu)A(pu) ≥ 0, (6.31)
which is manifestly non-negative.
6.4 Relaxing the conditions on ∆φ
The conditions (6.21) are stronger than necessary because we have not assumed anything
about the quantity that K∆,J(x1, x2) is integrated against. We can somewhat relax them
as follows. Note that poles in i〈VO∆,J(−∞z, z)V 〉 come from the region where x1, x2 are
near the lightray Rz. In this region, we expect the correlator 〈Ω|[V , φ(x1)][φ(x2), V ]|Ω〉 to
depend most strongly on the positions v1, v2 of the operators along the light-ray and simple
invariants built out of the relative position x1 − x2, since V, V are far from the light ray.
To be more precise, consider the integral over x1, x2 in the coordinates of section 3.4,
2J iµ(∆,J)SE(φφ[O])L(φφ[O])
(〈φφO〉,〈φφO〉)E
× 1
4
∫dr
rdv1dv2dαd
d−2w1dd−2w2
2J−1v−1−∆−∆1−∆2+J
221
(α(1−α)+(1−α)w2
1+αw22
)1−∆
(1+w212)
∆1+∆2+J−∆2 α
∆1−∆2+2−∆−J2 (1−α)
∆2−∆1+2−∆−J2
×r−∆−∆1−∆2−J
2 φ(−rα,v1,(rv21)12 w1)φ(r(1−α),v2,(rv21)
12 w2). (6.32)
The most important quantities built from x12 are
v21, x212 = rv21(1 + w2
−). (6.33)
Let us make the approximation that, to leading order in r, the correlator 〈[V , φ][φ, V ]〉depends only on v1, v2 and x2
12. That is, let us replace
φ(−rα,v1,(rv21)12 w1)φ(r(1−α),v2,(rv21)
12 w2)∼φ
(− r
2(1+w2
−),v1,0)φ(r
2(1+w2
−),v2,0).
(6.34)
This approximation would be valid, for example, if we could perform the OPE φ(x1)×φ(x2),
since the leading terms in the OPE depend only on v21 and x212. However, our assumption
is weaker than assuming that we can perform the OPE.
After rescaling r → r/(1 + w2−), we can now perform the integrals over α and w±,
following the methods in appendix D. The result is
i〈VO+∆,J(−∞z, z)V 〉
∼ 2d+J−4
π
∫dr
rdv1dv2 r
2∆φ−∆+J
2 v2∆φ−∆−J−2
221 〈Ω|[V , φ(− r
2 , v1, 0)][φ( r2 , v2, 0), V ]|Ω〉
= − 2d+J−4
πΓ(
∆+J+2−2∆φ
2
) ∫ dr
rr
2∆φ−∆+J
2
∫ ∞0
dk k∆+J−2∆φ
2 〈Ω|Θk(r)Θk(r)|Ω〉, (6.35)
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JHEP11(2018)102
where
Θk(r) ≡∫ ∞
0dv e−kv[φ( r2 , v, 0), V ]. (6.36)
The integrand in (6.35) should be correct to leading order at small r, which means the
leading residue of i〈VO∆,J(−∞z, z)V 〉 should be correct. This residue is manifestly posi-
tive whenever
∆φ <∆ + J + 2
2. (6.37)
For example, this proves the continuous spin ANEC for all J ≥ 2 if the lowest-dimension
scalar in the theory has dimension ∆φ ≤ d+42 .
7 Discussion
We have argued that every CFT contains light-ray operators that provide an analytic
continuation in spin of the light-transforms of local operators. This gives a physical in-
terpretation of Caron-Huot’s Lorentzian inversion formula [16]. Our construction involves
smearing two primary operators O1,O2 against a kernel to produce an object O∆,J , and
then taking residues in ∆ to localize the operators along a null ray. We have not shown
rigorously that the integral localizes to a null ray (as opposed to a lightcone). However, we
expect this is true based on the example of MFT and the fact that it’s true for integer J .
More generally, we expect that any singularity in the (∆, J)-plane should lead to a light-ray
operator. (For instance, one could take the discontinuity across a branch cut instead of a
residue.) It would be nice to understand better the structure of the (∆, J)-plane in general
CFTs. We know that for nonnegative integer J , the object O∆,J has simple poles in ∆ at
the locations of local operator dimensions. However, we do not know how it behaves for
general complex J .61 We also have not addressed the question of whether different oper-
ators O1,O2 produce different light-ray operators. We expect that in a nonperturbative
theory, the same set of light-ray operators should appear in every product OiOj , if allowed
by symmetry. It would be nice to show this rigorously.
Light-ray operators have the advantage over local operators that they fit into a more
rigid structure, due to analyticity in spin. However, unlike local operators, they are not
included in the Hilbert space of the CFT on Sd−1 because they annihilate the vacuum. One
way to realize them as states is to double the Hilbert space (with time running forwards in
one copy and backwards in the other). The Oi,J then become states in the doubled Hilbert
space.62 A general message is that the doubled Hilbert space contains interesting structure
that is not visible in a single copy, and it would be interesting to explore this idea further.
We have seen that light-ray operators enter the Regge limit of CFT four-point func-
tions. It would be nice to understand the actual spectrum and OPE coefficients of
61In planar N = 4 SYM, beautiful pictures of the (∆, J)-plane have been constructed using integrabil-
ity [83–86].62Oi,J itself is a somewhat violent state. However, we can regularize it by acting on the thermofield
double state with some temperature β. We thank Alexei Kitaev for this suggestion.
– 68 –
JHEP11(2018)102
continuous-spin light-ray operators in important physical theories (e.g. the 3d Ising model,
N = 4 SYM, and more), in order to determine what the Regge limit actually looks like in
those theories.63 Such operators have been explored in weakly-coupled gauge theories (see
e.g. [35–40]), and it would be interesting to study other perturbative examples. For exam-
ple, can one write a continuous-spin generalization of the Hamiltonian of the Wilson-Fisher
theory [87]?
Another important question is the extent to which light-ray operators form a complete
basis for describing the Regge regime. Indeed, in our discussion in section 5, we ignored
certain non-growing contributions in the Regge limit. It would be interesting to include
them and give them operator interpretations. Perhaps lightcone operators or other types
of nonlocal operators play a role. This question is also interesting in 1 dimension, where
the analog of the Regge regime is the so-called “chaos regime” of a four-point function.
In any spacetime dimension, we can ask: is there a complete basis of nonlocal operators
transforming as primaries in Lorentzian signature? Identifying a complete basis could help
in developing a generalization of the OPE that is valid in non-vacuum states. (The usual
OPE still works as an asymptotic expansion in non-vacuum states, but we would like to
find a convergent expansion.) Such a generalization would be a powerful tool for studying
Lorentzian physics.
Relatedly, it would be interesting to study OPEs of light-ray operators with each
other, especially the ANEC operator E = L[T ].64 In “conformal collider physics” [41] one
considers ANEC operators starting at the same point E(x, z1)E(x, z2) (usually taken to be
spatial infinity x =∞, so that the light-rays lie along future null infinity), and it is natural
to study the limit where their polarization vectors coincide z1 → z2. This question was
explored in [41], where it was argued that the leading term in the E×E OPE in N = 4 SYM
is a particular spin-3 light-ray operator that can be described in bulk string theory using
the Pomeron vertex operator of [8]. It would be nice to determine a systematic expansion
for this limit in a general CFT. Such an expansion could be useful for computing energy
correlators and studying jet substructure in CFTs. Light-ray operators could also be useful
for understanding aspects of deep inelastic scattering and PDFs.65
In this work, inspired by Caron-Huot’s beautiful result [16], we have been led to an
unusual hybrid of Euclidean and Lorentzian harmonic analysis, i.e. harmonic analysis with
respect to the groups SO(d + 1, 1) and SO(d, 2). However, many of the resulting for-
mulae suggest that it might be fruitful to start with SO(d, 2) from the beginning. For
example, after applying the Sommerfeld-Watson trick, Regge correlators are written as
an integral over ∆ and J , which is suggestive of an expansion in Lorentzian principal
series representations (this observation was also made recently in [88]). It will be impor-
tant to develop this area further and explore its implications for many of the above ques-
tions.66
63Besides planar N = 4 SYM, another CFT where the Regge limit of a four-point function has been
computed is the 2d (supersymmetric) SYK model [14].64We thank Sasha Zhiboedov for discussion on this point.65We thank Juan Maldacena for this suggestion.66We thank Abhijit Gadde for emphasizing this idea.
– 69 –
JHEP11(2018)102
The intrinsically Lorentzian integral transforms introduced in section 2.3 have been
a key computational tool in this work. These transforms have a natural group-theoretic
origin as Knapp-Stein intertwining operators for SO(d, 2), but they can also be applied to
representations of SO(d, 2). In this work, we have focused primarily on the light-transform,
but the remaining transforms may also have interesting applications. For example, it would
be interesting to compute the full monodromy matrix for spinning conformal blocks in
terms of intertwining operators, generalizing (5.23). Steps in this direction have already
been taken in [60].
One concrete result of this work is a generalization of Caron-Huot’s Lorentzian in-
version formula to four-point correlators of operators in arbitrary Lorentz representations.
Caron-Huot’s original formula has already proven useful in a variety of contexts [89–95],67
and we hope that our generalization will be similarly useful. For example, one might try
to determine all four-point functions in theories with weakly-broken higher spin symmetry,
generalizing the results of [93]. It would also be interesting to study inversion formulae
in the context of stress-tensor four-point functions, perhaps making contact with the sum
rules in [43, 98].
An important application of Lorentzian inversion formulae is to the lightcone bootstrap
and large-spin perturbation theory [17–26, 79]. Lorentzian inversion formulae make it
particularly simple to study OPE coefficients and anomalous dimensions of “double-twist
operators” [17, 18] and averaged OPE data for “multi-twist” operators (see e.g. [94, 95]).
An important problem for the future is to disentangle individual multi-twist trajectories.
It is likely that this will require studying crossing symmetry for higher-point functions. We
hope that light-ray operators will offer a useful perspective on this problem.
Another result of this work is a new proof of the average null energy condition (ANEC),
obtained by combining the causality-based proof of [46] with the idea of an inversion
formula. Our proof has some technical advantages over [46]. For example, it does not
use the OPE outside its regime of validity, and it also allows one to move away from
the asymptotic lightcone limit. However, it also has disadvantages. In particular, our
proof requires the CFT to contain a sufficiently low-dimension operator, and this condition
is absent in [46]. It would be interesting to understand whether this condition can be
relaxed further while still using an inversion formula. Another technical point that is
worth clarifying is the role/necessity of Rindler positivity, as opposed to the more easily-
established “wedge reflection positivity” [78] or the traditional positivity of norms.
The ANEC has a growing list of interesting applications in conformal field theory [41,
43, 44, 99–102]. However its higher-spin generalizations [46] have been less well-explored.
We have additionally proven that the ANEC holds for continuous spin — i.e. on the entire
leading Regge trajectory. It would be interesting to understand the implications of this
result, for example in a holographic context. (See [103] for recent work on shockwave oper-
ators, which are holographically dual to light-ray operators.) It would also be interesting
to understand the information-theoretic role of continuous-spin operators. How do they
67See also [96, 97] for applications of Lorentzian inversion formulae to quantities other than vacuum four-
point functions. It would be interesting to understand whether light-ray operators offer a useful perspective
on these works.
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JHEP11(2018)102
behave under modular flow? Can they appear in OPEs of entangling twist defects? The
ANEC can be improved to the quantum null energy condition (QNEC) [104, 105], which
was recently proven in [106] together with a higher integer spin generalization. Is there a
continuous-spin version of the QNEC?
Acknowledgments
We thank Clay Cordova, Thomas Dumitrescu, Abhijit Gadde, Luca Iliesiu, Daniel Jafferis,
Alexei Kitaev, Murat Kologlu, Raghu Mahajan, Eric Perlmutter, Matt Strassler, and Aron
Wall for helpful discussions. We thank Simon Caron-Huot, Tom Hartman, Denis Karateev,
Juan Maldacena, Douglas Stanford, and Sasha Zhiboedov for discussions and comments on
the draft. DSD is supported by Simons Foundation grant 488657 (Simons Collaboration on
the Nonperturbative Bootstrap). This work was supported by DOE grant DE-SC0011632
and the Walter Burke Institute for Theoretical Physics.
A Correlators and tensor structures with continuous spin
In this appendix we assume that there exists a continuous-spin operator O(x, z) and study
its Wightman functions. Note that here we are concerned with physical correlators. In
other parts of this paper we discuss the existence of continuous-spin conformal invari-
ants for fixed causal relations between the operator insertions, which is a very different
problem — Wightman functions must be well defined for arbitrary causal relationships
between points.
A.1 Analyticity properties of Wightman functions
Recall that Wightman functions of local operators are analytic in their arguments when the
appropriate iε prescription is introduced. More precisely, consider a Wightman function of
local operators (suppressing polarization vectors for simplicity)
〈Ω|On(xn) · · · O1(x1)|Ω〉, (A.1)
and let us split each xk into its real and imaginary parts,
xk = yk + iζk, yk, ζk ∈ Rd−1,1. (A.2)
The Wightman function (A.1) is analytic in the following region [65, 107] (see [11] for a
nice review):68
ζ1 > ζ2 > · · · > ζn. (A.3)
Here, the notation p > q means that p− q is timelike and future-pointing. We will refer to
this analyticity property as positive-energy analyticity.
68In fact, these functions are analytic in an even larger region [65, 107], but we do not study consequences
of this extended analyticity in this work.
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JHEP11(2018)102
Positive-energy analyticity can be derived in the following way. We first represent the
Wightman function (A.1) as a Fourier transform
〈Ω|On(xn) · · · O1(x1)|Ω〉 =
∫ddp1
(2π)d· · · d
dpn(2π)d
e−ip1x1...−ipnxn〈Ω|On(pn) · · · O1(p1)|Ω〉.
(A.4)
The existence of the Fourier transform follows from the Wightman temperedness axiom.
The Heisenberg equation implies
[H,Oi(xi)] = −i ∂∂x0
i
Oi(xi) =⇒ [H,Oi(pi)] = p0iOi(pi), (A.5)
and thus
HOi(pi) · · · O1(p1)|Ω〉 = (p01 + . . .+ p0
i )Oi(pi) · · · O1(p1)|Ω〉. (A.6)
In physical theories, all states have positive energies. Furthermore, positivity should hold
in any Lorentz frame. Thus, we conclude that whenever 〈Ω|On(pn) · · · O1(p1)|Ω〉 is nonva-
nishing,
p1 + . . .+ pi ≥ 0 (i = 1, . . . , n). (A.7)
Here, the notation p ≥ 0 means that p is timelike or null and future-pointing. Note that
the real part of the exponential factor in (A.4) is given by
exp (ζ1 · p1 + . . . ζn · pn) = exp[(pn + . . .+ p1) · ζn+ (pn−1 + . . .+ p1) · (ζn−1 − ζn)
+ (pn−2 + . . .+ p1) · (ζn−2 − ζn−1)
+ . . .
+ p1 · (ζ1 − ζ2)], (A.8)
where ζk = Im(xk). By translation-invariance, the first term in the exponential (pn + . . .+
p1) ·ζn can be replaced with zero. Suppose that the ζk satisfy (A.3). Due to (A.7), all other
terms in the exponential are non-positive and serve to damp the integral (A.4). Thus, we
can make sense of the Wightman function as an analytic function in this region.
The above discussion in no way depends on locality properties of Oi. The only in-
formation about Oi that we needed was the Heisenberg equation (A.5). This is of course
also satisfied by continuous-spin primary operators O(x, z), because it is simply part of
the definition of being primary. This means that positive-energy analyticity also holds
for Wightman functions involving continuous-spin operators. In the main text we con-
struct examples of continuous-spin operators for which positive-energy analyticity can be
checked explicitly.
This clarifies the properties of O(x, z) with respect to x. However, O(x, z) is also a
non-trivial function of z, and it is interesting to study analyticity in z. For this, assume that
we have already adopted the appropriate iε-prescription. By using Lorentz and translation
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JHEP11(2018)102
symmetries we can assume that we have inserted O(x, z) at x = iεe0 = (iε, 0, . . . , 0) with
ε > 0. Then we have for i, j = 1 . . . d− 1
[Mij ,O(iεe0, z)] = (zj∂zi − zi∂zj )O(iεe0, z), (A.9)
and so we have an Spin(d−1) ⊂ SO(d, 2) subgroup which stabilizes position of O and allows
us to change z. In particular, together with the homogeneity property (2.21) it allows us
to relate all future-directed null z to z = e0 + e1 = (1, 1, 0, . . . , 0). Let Uz ∈ Spin(d − 1)
that takes αz(e0 + e1) with αz > 0 to z. Then for a Wightman function with a single
continuous-spin operator we can write
〈Ω|On(xn) · · · Ok(xk)O(iεe0, z)Ok−1(xk−1) · · · O1(x1)|Ω〉 =
= αJz 〈Ω|On(xn) · · · Ok(xk)UzO(iεe0, e0 + e1)U †zOk−1(xk−1) · · · O1(x1)|Ω〉, (A.10)
and compute the right hand side by acting with Uz and U †z on the left and on the right.
This action will act on the spin indices of local operators and also shift their positions.
Change in the positions will, however, preserve the ordering of imaginary parts ζk (A.2),
and thus the Wightman function will remain in the region of analyticity.69 Since we can
take Uz to depend on z analytically in a neighborhood of any given z, this implies that
in the absence of other continuous-spin operators the left hand side of (A.10) should be
analytic in z.
It would be interesting to understand the analyticity conditions in z in presence of
other continuous spin operators. This might depend on some extra assumptions about the
nature of such operators, but it is natural to expect them to still be analytic. At least this
is the case for the integral transforms defined in section 2.3, since at fixed iε-prescription
these involve integrals of analytic functions.
A.2 Two- and three-point functions
Let us now study examples of Wightman functions of continuous-spin operators from the
point of view of positive-energy analyticity. This is especially interesting in CFTs because
the analytic structure of two- and three-point functions is fixed by conformal symmetry, and
this turns out to be in strong tension with positive-energy analyticity. For simplicity, we
focus on correlation functions involving the minimal number of continuous-spin operators.
We also restrict to traceless-symmetric tensor operators. However, the same statements
hold for general representations because the part of the tensor structure responsible for the
discrete spin labels λ is always positive-energy analytic.
A conformally-invariant two-point function of traceless-symmetric operators has
the form
〈O(x1, z1)O(x2, z2)〉 ∝(2(x12 · z1)(x12 · z2)− x2
12(z1 · z2))J
x2(∆+J)12
. (A.11)
69Note that in principle the stabilizer of iεe0 includes a full Spin(d) ∈ SO(d, 2). However, some of the
transformations in Spin(d)\Spin(d − 1) will change ordering of ζk and thus move Wightman function out
of the region of analyticity.
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JHEP11(2018)102
It is easy to check that the denominator is positive-energy analytic for any choice of
Wightman ordering, and we only need to study the numerator. For generic z1 and z2
we can write
x12 = αz1 + βz2 + x⊥, (A.12)
where x⊥ · zi = 0. Note that x⊥ is spacelike, because it is orthogonal to the timelike vector
z1 + z2. (Recall that all polarization vectors are null and future-directed.) The numerator
then takes the form(2(x12 · z1)(x12 · z2)− x2
12(z1 · z2))J
= (−z1 · z2)Jx2J⊥ > 0. (A.13)
On the one hand, we see that this is positive and well-defined for all real xi and zi. On the
other hand, we can show that it is only positive-energy analytic for integer J ≥ 0. Indeed,
selecting a Wightman ordering and adding appropriate imaginary parts as in (A.2), in any
case we find that ζ⊥ is a spacelike vector (we can make it non-zero), because it is orthogonal
to z1 + z2. This means that by choosing an appropriate y12 we can achieve
x2⊥ = y2
⊥ − ζ2⊥ + 2i(y⊥ · ζ⊥) = 0, (A.14)
and in particular wind x2⊥ around zero without leaving the region of positive-energy ana-
lyticity.70,71 Thus (A.13) can not be analytic there unless J is a non-negative integer.
This implies that the only way the Wightman two-point function of a generic contin-
uous spin operator O can be positive-energy analytic is by being zero,72
〈Ω|O(x1, z1)O(x2, z2)|Ω〉 = 0. (A.15)
In unitary theories vanishing of this two-point function implies
O(x, z)|Ω〉 = 0. (A.16)
This gives another derivation of the fact stated in the introduction: continuous-spin oper-
ators must annihilate the vacuum.
Let us now consider a three-point function with a single continuous-spin operator O,
〈O1(x1, z1)O2(x2, z2)O(x3, z3)〉 ∝ f(xi, zi)
(x13 · z3
x213
− x23 · z3
x223
)J3−n3
, (A.17)
70To be specific, we can wind x2⊥ around 0 once with y12 returning to the original position, and thus
for (A.13) to be single-valued, we need J ∈ Z.71This argument doesn’t work in d = 3 because then y⊥ and ζ⊥ are forced to lie in the same 1-dimensional
subspace. In that case we are still free to change both y⊥ and ζ⊥, and thus x⊥ = y⊥+iζ⊥, in a neighborhood
of 0. This leads to a weaker requirement that J ∈ 12Z≥0. This has to do with the fact that for d = 3 the null-
cone is not simply-connected and it makes sense to consider multi-valued functions of z. In fact, fermionic
operators can be described by double-valued functions of z. (If we write zµ = χαχβσαβµ for a real spinor
χ, then we get polynomial functions of χ.) Our argument thus shows that only single- and double-valued
functions of z are consistent with positive-energy analyticity. In higher dimensions we cannot describe
fermionic representations by using a single null polarization and thus we do not get this subtlety.72We derived this for generic z1 and z2, but as discussed in the previous section, we expect the Wightman
functions to be continuous in polarizations.
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JHEP11(2018)102
where f(xi, zi) is the part of the tensor structure which is manifestly positive-energy ana-
lytic, and is a homogeneous polynomial in z3 with degree n3 ≥ 0. The non-trivial part of
the correlator can be written as(x13 · z3
x213
− x23 · z3
x223
)J3−n3
= (v12,3 · z3)J3−n3 , (A.18)
where
v212,3 =
(x13
x213
− x23
x223
)2
=x2
12
x213x
223
. (A.19)
We see that v12,3 can be both spacelike and timelike, depending on the causal relationship
between the three points xi. This immediately implies that, for example, when all xijare spacelike, the inner product v12,3 · z3 is not sign-definite and we need to invoke iε-
prescriptions to define (v12,3 · z3)J3−n3 , even for purely Euclidean configurations. For the
iε-prescriptions to make sense, the tensor structure must be positive-energy analytic. This
means that in this situation, positive-energy analyticity is not only required for correlators
to make physical sense, but also simply for the tensor structures to be single-valued.73 To
proceed, note that in the region of positive-energy analyticity x2ij 6= 0 and furthermore
the map
x 7→ x
x2(A.20)
preserves the set of x = y + iζ with future-directed (past-directed) timelike ζ.74 Since
it is also its own inverse, this implies that by varying x13 and x23 within the region of
positive-energy analyticity, we can reproduce any pair of values for q1 = x13
x213
and q2 = x23
x223
with imaginary parts satisfying the same constraints as those of x13 and x23 respectively.
This means that in the region of positive-energy analyticity for the orderings
〈0|O2OO1|0〉 and 〈0|O1OO2|0〉, (A.21)
the vector v12,3 = q1 − q2 has a timelike imaginary part restricted to be future-directed or
past-directed respectively, while for the orderings
〈0|OiOjO|0〉 and 〈0|OOiOj |0〉 (A.22)
this imaginary part is not restricted at all. In the former case v12,3 · z3 has either negative
or positive imaginary part, and thus the inner product cannot vanish or wind around zero,
while in the latter case this inner product can vanish or wind around zero. We thus conclude
73This is in contrast to the two-point Wightman function case considered above, where (A.13) is single-
valued without the iε-prescription.74If x2 = (y + iζ)2 = y2 − ζ2 + 2iy · ζ = 0 with timelike ζ, then y · ζ = 0, which implies that y is
spacelike and thus y2 − ζ2 > 0, leading to contradiction. Imaginary part of xx2 is, up to a positive factor,
ζ(y2 − ζ2)− 2y(y · ζ). For y = 0 this is timelike and has the same direction as ζ. For any y, this squares to
ζ2((y2 − ζ2)2 + 4(y · z)2) < 0, and thus by continuity Im xx2 remains timelike in the direction of ζ.
– 75 –
JHEP11(2018)102
that the Wightman functions (A.21) are positive-energy analytic for any value of J3, while
the Wightman functions (A.22) are positive-energy analytic only for integer J3 ≥ n3.75
Again, recalling that the physical Wightman functions of continuous-spin opera-
tors must be positive-energy analytic, we are forced to conclude that Wightman func-
tions (A.22) vanish,
〈Ω|O1O2O|Ω〉 = 〈Ω|OO1O2|Ω〉 = 0, (A.23)
which of course consistent with the fact that O annihilates the vacuum. An interesting
observation is that the distinction we made above between the Wightman orderings (A.21)
and (A.22) conflicts with microcausality, because for spacelike-separated points all these
Wightman functions would be equal.76 This means that non-trivial continuous-spin op-
erators must be non-local, as stated in the introduction, in the sense that they cannot
satisfy microcausality.
A consequence of non-locality is that a physical correlator involving a continuous-spin
operator is not well-defined without specifying an operator ordering even if all the distances
are spacelike. This in particular means that time-ordered correlators are not quite well-
defined in the presence of continuous-spin operators (i.e. how do we order O when it is
spacelike from something?). This also makes it unclear how one would define Euclidean
correlators for continuous spin (the usual Wick-rotation to Euclidean signature requires
micro-causality). Another problem with attempting to describe continuous-spin operators
in Euclidean signature is that under Euclidean rotation group SO(d) the orbit of a single
null direction in Rd−1,1 consists of all null directions in Cd. Thus we would need to define
O(x, z) for all complex null z, but above it was very important to have future-directed real
z to establish positive-energy analyticity of at least some Wightman functions.
A.3 Conventions for two- and three-point tensor structures
When working with integer spin the simplest way to specify standard tensor structures is
to give their expressions in Euclidean signature or, equivalently, in Lorentzian signature
with all points are spacelike separated. With continuous spin, Euclidean signature is not
an option, and as we saw above even for spacelike separations in Lorentzian signature care
must be taken to define phases of three-point functions. In this section we briefly record
our conventions for symmetric tensor operators.
We will choose the following convention for a two-point function in Lorentzian signa-
ture:
〈O(x1, z1)O(x2, z2)〉 =(−2z1 · I(x12)z2)J
x2∆12
Iµν(x) = δµν − 2xµxνx2
. (A.24)
75Recall that n3 ≤ J3 is the standard condition that we encounter when dealing with integer-spin tensor
structures, it just means that f(xi, zi) must be a polynomial in z3 of degree at most J3. The 3d subtlety
we discussed in footnote 71 would be visible here as well, if we allowed f to be double-valued in z (and
polynomial in χ), which would correspond to making the product O1O2 fermionic, thus forcing J to be
half-integer.76Recall that as noted above, the region of spacelike separation is the problematic one, because there
v12,3 is spacelike and v12,3 · z3 is not sign-definite.
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JHEP11(2018)102
The nonstandard numerator is so that the two-point function is positive when 1 and
2 are spacelike separated and z1,2 are future-pointing null vectors. For local operators
this completely defines standard Wightman two-point functions via iε prescriptions. For
continuous-spin operators physical Wightman functions vanish, but we still need two-point
conformal invariants in some calculations (like the definition of the S-transform), and for
these purposes it suffices to specify the two-point invariant for spacelike x12.
Now consider a three-point function 〈φ1(x1)φ2(x2)O(x3, z)〉, where φ1 and φ2 are
scalars and O has dimension ∆ and spin J . We demand that the correlator (either
Wightman or time-ordered) should be positive when 1, 2, 3 are mutually spacelike and
z · x23 x213 − z · x13 x
223 > 0. Our precise convention is
〈φ1(x1)φ2(x2)O(x3, z)〉 =
(2z · x23 x
213 − 2z · x13 x
223
)Jx∆1+∆2−∆+J
12 x∆1+∆−∆2+J13 x∆2+∆−∆1+J
23
. (A.25)
This is unambiguous for local operators since at spacelike separations there is no differ-
ence between various Wightman orderings and time-ordering.77 If J is continuous, we are
necessarily talking about a Wightman function and we need to specify the ordering. Our
choice is
〈0|φ1(x1)O(x3, z)φ2(x2)|0〉 =
(2z · x23 x
213 − 2z · x13 x
223
)Jx∆1+∆2−∆+J
12 x∆1+∆−∆2+J13 x∆2+∆−∆1+J
23
, (A.26)
defined to be positive under the same conditions as (A.25).
The nontraditional factors of 2 in (A.24) and (A.25) are so that the associated confor-
mal blocks have simple behavior in the limit of small cross-ratios
〈φ1φ2O〉〈Oφ3φ4〉〈OO〉
∼(∏
x#ij
)χ
∆−J2 χ
∆+J2 χ χ 1. (A.27)
They also simplify several formulae in the main text.
B Relations between integral transforms
B.1 Square of light transform
In this appendix we explicitly compute the square of the light transform. In order to do
this, we need to assume that the operator that the light transform acts upon belongs to
the Lorentzian principal series
∆ =d
2+ is, J = −d− 2
2+ iq, (B.1)
so that ∆ + J = 1 + i(s + q) = 1 + iω and ∆L + JL = 2−∆− J = 1− i(s + q) = 1 − iωand thus both the first and the second light transforms make sense if w 6= 0.
77Note however that this notation for the standard structure is somewhat abusive. For physical correlators
we of course have 〈φ1φ2O〉Ω = 〈φ2φ1O〉Ω, but the standard structure (A.25) gains a (−1)J under this
permutation. This leads to several appearances of (−1)J in our formulas which are awkward to explain.
– 77 –
JHEP11(2018)102
It will also be convenient to use the expression for the light transform in the coordinates
(τ,~e) on Md. In these coordinates the polarization vector z can be described as (z0, ~z)
where ~z is tangent to Sd−1 at ~e, i.e. ~z · ~e = 0, and we have (z0)2 = |~z|2. We then have
L[O](τ,~e;z0,~z) =
∫ π
0dκ(sinκ)∆+J−2(z0)1−∆O(τ+κ,cosκ~e+sinκ ~z
z0 ;1,cosκ ~zz0−sinκ~e).
(B.2)
Note that this form also makes it manifest that there is no singularity associated to α = 0
in (2.37).
The square of light transform becomes
L2[O](τ,~e;z0,~z) =
∫ π
0
∫ π
0dκdκ′(z0)J(sinκ′)−∆−J(sinκ)∆+J−2
×O(τ+κ+κ′,cos(κ+κ′)~e+sin(κ+κ′) ~zz0 ;1,cos(κ+κ′) ~z
z0−sin(κ+κ′)~e)
=
∫ 2π
0dκK(κ)(z0)JO(τ+κ,cosκ~e+sinκ ~z
z0 ;1,cosκ ~zz0−sinκ~e), (B.3)
where
K(κ) =
∫ min(κ/2,π−κ/2)
max(−κ/2,κ/2−π)dη(sin
κ
2− η)−1−iω(sin
κ
2+ η)−1+iω. (B.4)
To compute K(κ), for κ 6= 0, π, 2π we can use the substitution
eβ =sin(κ2 + η
)sin(κ2 − η
) , (B.5)
which turns the integral into
K(κ) =1
sinκ
∫ +∞
−∞dβeiwβ = 0, (ω 6= 0). (B.6)
This means that K(κ) is supported at κ = 0, π, 2π. Let us thus consider first the con-
tribution near κ = 0. Near κ = 0 we can expand both sines and find, introducing a
regulator ε,
K(κ) =
∫ κ/2
−κ/2dη(κ
2− η)−1−iω+ε (κ
2+ η)−1+iω+ε
= κ−1+2ε
∫ 1/2
−1/2dη
(1
2− η)−1−iω+ε(1
2+ η
)−1+iω+ε
= (2ε)κ−1+2εΓ(iω + ε)Γ(−iω + ε)
(2ε)Γ(2ε). (κ 1) (B.7)
For ε→ 0, using
(2ε)κ−1+2ε → δ(κ), (κ > 0) (B.8)
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JHEP11(2018)102
we find
K(κ) = Γ(−iω)Γ(iω)δ(κ) =π
(∆ + J − 1) sinπ(∆ + J)δ(κ), (κ 1). (B.9)
The calculation near κ = 2π is the same and thus we have
K(κ) =π
(∆ + J − 1) sinπ(∆ + J)(δ(κ) + δ(κ− 2π)) + 〈contribution from π〉 (B.10)
To find the contribution from κ = π, write κ = π − r for small 0 < r 1.78 We
have now
K(κ) =
∫ π2− r
2
−π2
+ r2
dη(
sinπ
2− r
2−η)−1−iω (
sinπ
2− r
2+η)−1+iω
=
∫ π−r
0dη (sinr+η)−1−iω (sinη)−1+iω
≈∫ Nr
0dη (r+η)−1−iω+ε η−1+iω+ε+
∫ Nr
0dη (r+η)−1+iω+ε η−1−iω+ε
= r−1+2ε
[∫ ∞0
dη (1+η)−1+iω+ε η−1−iω+ε+
∫ ∞0
dη (1+η)−1−iω+ε η−1+iω+ε
]= r−1+2ε πΓ(1−2ε)
Γ(2−J−∆−ε)Γ(J+∆−ε)(csc(π(J+∆−ε))−csc(π(J+∆+ε))) . (B.11)
Here 0 r Nr 1 and the two terms come from the two sides of the integral. We can
now compute for small Λ > 0
limε→0
∫ π
π−ΛK(κ)dκ = − π cosπ(∆ + J)
(∆ + J − 1) sinπ(∆ + J). (B.12)
Recalling also that there is also a contribution from the negative values of r, we find the
final result
K(κ) =π
(∆ + J − 1) sinπ(∆ + J)(δ(κ)− 2 cosπ(∆ + J)δ(κ− π) + δ(κ− 2π)) . (B.13)
In terms of action on O this immediately implies
L2 =π
(∆ + J − 1) sinπ(∆ + J)
(1− 2 cosπ(∆ + J)T + T 2
)=
π
(∆ + J − 1) sinπ(∆ + J)
(T − eiπ(∆+J)
)(T − e−iπ(∆+J)
). (B.14)
B.2 Relation between shadow transform and light transform
In this appendix we prove the relation (2.87). As in the preceding part of this appendix,
we must assume that (2.87) acts on an operator in the Lorentzian principal series so that
this action is well-defined. We have
LSJL[O](x, z) =
∫Dd−2z′dα1dα2(−α1)−∆−J(−α2)d−2+J−∆(−2z · z′)1−d+∆
×O(x− z′/α1 − z/α2, z′) (B.15)
78There is going a similar contribution from r < 0.
– 79 –
JHEP11(2018)102
Let us write x′ = x− z′/α1 − z/α2. Then we have
I(x− x′)z = z − 2(z′/α1 + z/α2)(z′/α1 + z/α2) · z
(z′/α1 + z/α2)2= −α2
α1z′. (B.16)
Considering the integral in the region of large negative α1 and α2 we find∫Dd−2z′dα1dα2(−α1)−∆−J(−α2)d−2+J−∆
(−α1α2(x− x′)2
)1−d+∆(α1
α2
)J×O(x′,−I(x− x′)z)
=
∫Dd−2z′dα1dα2(−α1)1−d(−α2)−1
(−(x− x′)2
)1−d+∆O(x′,−I(x− x′)z) (B.17)
We would now like to replace the integral∫Dd−2z′dα1dα2 by
∫ddx′. For this we write
1 =
∫ddx′δd(x− x′ − z′/α1 − z/α2) (B.18)
and then compute∫Dd−2z′dα1dα2(−α1)1−d(−α2)−1δd(x− x′ − z′/α1 − z/α2)
=
∫ddz′dα1dα2
volRθ(z′0)δ(z′2)(−α1)(−α2)−1δd(−α1(x− x′) + z′ + α1z/α2)
=
∫dα1dα2
volRδ(((x− x′)− z/α2)2)(−α1)−1(−α2)−1
= −(x− x′)−2. (B.19)
We thus conclude that (B.17) is equal to∫ddx′(−(x− x′)2)∆−dO(x′,−I(x− x′)z). (B.20)
More precisely, it is the contribution to (B.15) from the region of large negative αi. We
recognize that it has precisely the form of T -shifted Lorentzian shadow integral (2.31), i.e.
S∆ = iT −1LSJL. (B.21)
C Harmonic analysis for the Euclidean conformal group
C.1 Pairings between three-point structures
The conformal representation of an operator O is labeled by a scaling dimension ∆ and an
SO(d) representation ρ. The representation O† has dimension d−∆ and SO(d) represen-
tation ρ∗ (the dual of ρ). Thus, there is a natural conformally-invariant pairing between
n-point functions of Oi’s and n-point functions of O†i ’s, given by multiplying and integrating
over all points modulo the conformal group,(〈O1 · · · On〉, 〈O†1 · · · O
†n〉)E
=
∫ddx1 · · · ddxn
vol(SO(d+ 1, 1))〈O1 · · · On〉〈O†1 · · · O
†n〉. (C.1)
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JHEP11(2018)102
Here, we are implicitly contracting Lorentz indices between each pair Oi and O†i . The “E”
subscript stands for “Euclidean.”
This pairing is particularly simple for three-point structures. In that case, we can use
conformal transformations to set x1 = 0, x2 = e, x3 = ∞ (with e a unit vector), and no
integrations are necessary. The pairing becomes simply(〈O1O2O3〉, 〈O†1O
†2O†3〉)E
=1
2dvol(SO(d− 1))〈O1(0)O2(e)O3(∞)〉〈O†1(0)O†2(e)O†3(∞)〉.
(C.2)
The factor 2−d comes from the Fadeev-Popov determinant for the above gauge-fixing.79
The factor vol(SO(d− 1)) is the volume of the stabilizer group of three points.
As an example, a scalar-scalar-spin-J correlator has a single tensor structure
〈φ1φ2O3,J〉 given in (A.25). The pairing in that case is
(〈φ1φ2O3,J〉, 〈φ1φ2O3,J〉
)E
=22J
2dvol(SO(d− 1))(eµ1 · · · eµJ − traces)(eµ1 · · · eµJ − traces)
=22J CJ(1)
2dvol(SO(d− 1)), (C.3)
where CJ(x) is defined in (H.10).
C.2 Euclidean conformal integrals
Suppose O,O′ are principal series representations, with dimensions ∆ = d2 +is,∆′ = d
2 +is′
with s, s′ > 0 and SO(d) representations ρ, ρ′. A “bubble” integral of two three-point
functions is proportional to their three-point pairing,
∫ddx1d
dx2〈O1O2Oa(x)〉〈O†1O†2O′†b (x′)〉 =
(〈O1O2O〉, 〈O†1O
†2O†〉)E
µ(∆, ρ)δab δ(x− x′)δOO′ ,
δOO′ ≡ 2πδ(s− s′)δρρ′ . (C.4)
The right-hand side contains a term δOO′ restricting the representations O,O′ to be the
same, since this is the only possibility allowed by conformal invariance.80 Here, a, b are
indices for the representations ρ, ρ∗ of SO(d), respectively. We have suppressed the SO(d)
indices of the other operators, for brevity.
The factor µ(∆, ρ) in the denominator is called the Plancherel measure. It is known
in great generality [42] (see [68] for an elementary derivation). In this work, we will only
79Note that [31] used a convention where vol(SO(d+1, 1)) was defined to include an extra factor of 2−d to
cancel the Fadeev-Popov determinant. Here, we prefer not to cancel this factor because it simplifies other
formulae in this work.80Eq. (C.4) is sometimes written including two terms — one with δ(s − s′) and another with δ(s + s′).
Here we have only one term because we have restricted s, s′ > 0. The other term can be obtained by
performing the shadow transform on either O or O′†.
– 81 –
JHEP11(2018)102
need µ(∆, J) for symmetric traceless tensors:
µ(∆, J) =dim ρJ
2dvol(SO(d))
Γ(∆− 1)Γ(d−∆− 1)(∆ + J − 1)(d−∆ + J − 1)
πdΓ(∆− d2)Γ(d2 −∆)
,
dim ρJ =Γ(J + d− 2)(2J + d− 2)
Γ(J + 1)Γ(d− 1). (C.5)
Here dim ρJ is the dimension of the spin-J representation of SO(d).
Another conformal integral we will need is the Euclidean shadow transform of a three-
point function of two scalars and a symmetric traceless tensor
〈φ1φ2SE [O](y)〉 =
∫ddx〈O(y)O†(x)〉〈φ1φ2O(x)〉
= SE(φ1φ2[O])〈φ1φ2O(y)〉, (C.6)
where
SE(φ1φ2[O]) = (−2)Jπd/2Γ(∆− d
2)Γ(∆ + J − 1)
Γ(∆− 1)Γ(d−∆ + J)
Γ(d−∆+∆1−∆2+J2 )Γ(d−∆+∆2−∆1+J
2 )
Γ(∆+∆1−∆2+J2 )Γ(∆+∆2−∆1+J
2 ).
(C.7)
The factor of (−2)J relative to [31] is because we are using a different normalization con-
vention for the two-point function (A.24).
The square of the shadow transform is related to the Plancherel measure by [42] (see [68]
for an elementary derivation)
S2E =
1
µ(∆, ρ)
〈O(0)O†(∞)〉〈O(∞)O†(0)〉2dvol(SO(d))
≡ N (∆, ρ), (C.8)
where the indices in two-point functions are implicitly contracted. In the case of a spin-J
representation, we have
N (∆, J) =22J dim ρJ
2dµ(∆, J)vol(SO(d)). (C.9)
Indeed, we can easily verify
SE(φ1φ2[O])SE(φ1φ2[O]) = N (∆, J). (C.10)
C.3 Residues of Euclidean partial waves
In this section, we prove (3.8). The proof for primary four-point functions is standard (see
e.g. [31, 42]). We now give a slightly more complicated argument that works for n-point
functions. However, the key ingredients are identical to the standard argument.
Consider the integral in the completeness relation (3.3),
I =
∫ddxP∆,J(x)〈O(x)φ1φ2〉. (C.11)
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JHEP11(2018)102
The partial wave P∆,J also depends on the coordinates x3, . . . , xk, but they don’t play a
role in the current discussion so we have suppressed them. We have also suppressed Lorentz
indices. When we have a product of an operator and its shadow at coincident points, we
will assume their Lorentz indices are contracted.
Note that I is an eigenvector of the Casimirs of the conformal group acting simulta-
neously on points 1 and 2. Thus, it is completely determined by its behavior in the OPE
limit x1 → x2. There are two contributions in this limit. The first comes from the regime
where x is sufficiently far from x1, x2 that we can use the 1× 2 OPE inside the integrand:
〈φ1φ2O(x)〉 = C12O(x1, x2, x
′, ∂x′)〈O(x′)O(x)〉. (C.12)
Here, C12O is a differential operator that encodes the sum over descendants in the φ1 × φ2
OPE. The point x′ can be chosen arbitrarily inside a sphere separating x1, x2 from all
other points. We will abbreviate the right-hand side of (C.12) as C12O(x′)〈O(x′)O(x)〉.
Inserting (C.12) and applying the shadow transform to the definition of P∆,J (3.5), we find
I ⊃ C12O(x′)
∫ddx〈O(x′)O(x)〉P∆,J(x) = SE(φ1φ2[O])C
12O(x)P∆,J
(x). (C.13)
The second contribution to I comes from the regime where x is near both x1, x2 but
far away from all other points. In this case, we can insert a shadow transform and then
perform the OPE:
I = SE(φ1φ2[O])−1
∫ddxddx′P∆,J(x)〈O(x)O(x′)〉〈O(x′)φ1φ2〉
⊃ SE(φ1φ2[O])−1
∫ddxddx′P∆,J(x)〈O(x)O(x′)〉C12O(x′′)〈O(x′′)O(x′)〉
= SE(φ1φ2[O])−1N (∆, J)C12O(x)P∆,J(x)
= SE(φ1φ2[O])C12O(x)P∆,J(x). (C.14)
Where we have used (C.10).
The two contributions (C.13) and (C.14) are already eigenvectors of the conformal
Casimirs, so together they give the full answer for I. The two terms differ simply by the
replacement ∆↔ d−∆. Thus, we can plug them into the completeness relation (3.3) and
use ∆↔ d−∆ symmetry to extend the ∆ integral along the entire imaginary axis,
〈V3 · · ·VkO1O2〉Ω =∞∑J=0
∫ d2
+i∞
d2−i∞
d∆
2πiµ(∆, J)SE(φ1φ2[O])C12OP∆,J(x). (C.15)
Because C12O dies exponentially at large positive ∆, we can now close the ∆ contour to the
right and pick up poles along the positive real axis. Comparing to the physical operator
product expansion gives (3.8).
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JHEP11(2018)102
D Computation of R(∆1,∆2, J)
In this appendix we compute the coefficient R appearing in the first line of (3.39)
R(∆1,∆2,J)≡−2J−2
∫dαdd−2w1d
d−2w22J−1
(α(1−α)+(1−α)w2
1+αw22
)1−∆1−∆2−J
(1+w212)d−∆1−∆2α−∆1+1−J(1−α)−∆2+1−J .
(D.1)
As the first step, we do the wi integrals. We define w− = w12 and w+ = w1 + w2. The
integral over dwi becomes (without the −2J−2 and w-independent factors)
22(∆1+∆2+J)−d∫dd−2w+d
d−2w−
(4α(1−α)+w2
++w2−+2(1−2α)w+ ·w−
)1−∆1−∆2−J
(1+w2−)d−∆1−∆2
.
(D.2)
Now we shift w+ → w+ − (1− 2α)w− to find
22(∆1+∆2+J)−d∫dd−2w+d
d−2w−
(4α(1− α)(1 + w2
−) + w2+
)1−∆1−∆2−J
(1 + w2−)d−∆1−∆2
. (D.3)
Rescaling w+ we find∫dd−2w+d
d−2w−(α(1− α))1−∆1−∆2−J+ d−2
2(1 + w2
+
)1−∆1−∆2−J
(1 + w2−)J+ d
2
=
= (α(1− α))1−∆1−∆2−J+ d−22 × πd−2 Γ(J + 1)Γ(−d
2 + J + ∆1 + ∆2)
Γ(J + d2)Γ(J + ∆1 + ∆2 − 1)
. (D.4)
The remaining α-integral becomes∫dαα−∆2+ d−2
2 (1− α)−∆1+ d−22 =
Γ(d2 −∆1)Γ(d2 −∆2)
Γ(d−∆1 −∆2). (D.5)
Combining everything together we find
R(∆1,∆2, J) = −2J−2πd−2 Γ(J + 1)Γ(−d2 + J + ∆1 + ∆2)
Γ(J + d2)Γ(J + ∆1 + ∆2 − 1)
Γ(d2 −∆1)Γ(d2 −∆2)
Γ(d−∆1 −∆2). (D.6)
E Parings of continuous-spin structures
In this section we describe the natural conformally-invariant pairing between continuous
spin structures. Recall that the Euclidean pairings are constructed from the basic invari-
ant integral ∫ddxO(x)O†(x), (E.1)
where contraction of SO(d) indices is implicit. This integral is conformally-invariant be-
cause if O transforms in (∆, ρ) then O† transforms in (d−∆, ρ∗), where ρ∗ is the SO(d) irrep
– 84 –
JHEP11(2018)102
dual to ρ. We can therefore contract SO(d) indices and the dimensions in the integrand
add up to 0 (taking into account the measure ddx).
To pair continuous-spin structures in Lorentzian, we need to make use of the integral∫ddxDd−2zO(x, z)OS†(x, z) (E.2)
If O transforms in (∆, J, λ), then OS† transforms in (d−∆, 2− d− J, λ∗). The integrand
has 0 homogeneity in x and z, and λ-indices can be contracted.81
E.1 Two-point functions
Let us start with two-point functions. As discussed in section A, two-point functions of
continuous-spin operators do not make sense as Wightman functions, so in order to discuss
them, we have to think about them simply as some conformal invariants defined at least
for spacelike separated points.
That said, given a two-point structure for O in representation (∆, J, λ) and a two-point
function for OS in representation S[(∆, J, λ)] = (d − ∆, 2 − d − J, λ), we can define the
two-point pairing by
(〈OO†〉, 〈OSOS†〉)Lvol(SO(1, 1))2
≡∫x1≈x2
ddx1ddx2D
d−2z1Dd−2z2
vol(SO(d, 2))〈Oa(x1, z1)Ob†(x2, z2)〉〈OS
b (x2, z2)OS†a (x1, z1)〉, (E.3)
where factor vol(SO(1, 1))2 is for future convenience82 and the subscript “L” stands for
“Lorentzian.” On the right hand side, we divide by the volume of the conformal group
since the integral is invariant under it. Formally, this means that we should compute
the integral by gauge-fixing the action of conformal group and introducing an appropriate
Faddeev-Popov determinant. To perform gauge-fixing, we can first put x1 and x2 into some
standard configuration. A natural choice is to set x1 = 0 and x2 =∞ (spacelike infinity).83
This configuration is still invariant under dilatation and Lorentz transformations. Thus
81Given that OS transforms in (d−∆, 2− d− J, λ), it is a bit non-trivial to understand why OS† has λ∗.
In odd dimensions λ and λ∗ is the same irrep, so there is no question here. In even dimension † changes the
sign of the last row of Young diagram of (d−∆, 2−d−J, λ) in the same way as it does for all so(d)-weights.
In other words, it flips the sign if d = 4k and does nothing for d = 4k + 2. However, this last row is also
the last row of λ and λ is an SO(d − 2)-irrep. It then turns out that from the SO(d − 2) point of view,
this action is equivalent to taking the dual. Another way to see this is that † is complex conjugation for
SO(d − 1, 1), and thus for SO(d − 2), which can be thought of as a subgroup of SO(d − 1, 1). But since
SO(d− 2) is compact, for it complex conjugation is the same as taking the dual.82Similarly to the Euclidean case [68], the right hand side can be alternatively computed in terms of
Plancherel measure divided by vol(SO(1, 1))2. In Euclidean we get only one power of vol(SO(1, 1)), which
corresponds to the fact that there we have only one continuous parameter ∆, while in Lorentzian we have
both ∆ and J .83We define O(∞) = limL→∞ L2∆O(Le), where e is a conventional spacelike unit vector. We choose
e = (0, 1, 0, . . . , 0).
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JHEP11(2018)102
we have
(〈OO†〉,〈OSOS†〉)Lvol(SO(1,1))2 =
∫Dd−2z1D
d−2z2
2dvol(SO(1,1)×SO(d−1,1))〈Oa(0,z1)Ob†(∞,z2)〉〈OS
b (∞,z2)OS†a (0,z1)〉,
(E.4)
where 2d comes from the Faddeev-Popov determinant.84 If we define zR2 =
(z02 ,−z1
2 , z22 , . . . , z
d−12 ), so that Lorentz group transforms z1 and zR2 in the same way, the
integral ∫Dd−2z1D
d−2zR2vol(SO(d− 1, 1))
(E.5)
essentially becomes the (d− 2)-dimensional Euclidean conformal two-point integral. It can
also be computed by gauge-fixing, i.e. by setting zµ1 = zµ0 ≡ (12 ,
12 , 0, . . . , 0), which is the
embedding-space representation of the origin of Rd−2, zRµ2 = zµ∞ ≡ (12 ,−
12 , 0, . . . , 0), which
is the embedding-space representation of the infinity of Rd−2. The stabilizer group of this
configuration is SO(1, 1)×SO(d− 2), which consists of (d− 2)-dimensional dilatations and
rotations. We thus conclude
(〈OO†〉, 〈OSOS†〉)L =1
2d2d−2vol(SO(d− 2))〈Oa(0, z0)Ob†(∞, zR∞)〉〈OS
b (∞, zR∞)OS†a (0, z0)〉,
(E.6)
where we included another Faddeev-Popov determinant. Note that the right hand side is
proportional to dim λ.
We can summarize this result as follows. Note that the product
〈Oa(x1, z1)Ob†(x2, z2)〉〈OSb (x2, z2)OS†
a (x1, z1)〉 (E.7)
transforms in representation (∆, J, λ) = (d, 2−d, •) at both x1 and x2. Thus we must have
〈Oa(x1, z1)Ob†(x2, z2)〉〈OSb (x2, z2)OS†
a (x1, z1)〉 = A(−2z1 · I(x12)z2)2−d
x2d12
. (E.8)
For some constant A. Using (E.6), we find
(〈OO†〉, 〈OSOS†〉)L =A
22d−2vol(SO(d− 2)). (E.9)
E.2 Three-point pairings
We can analogously define a three-point pairing for continuous-spin structures,(〈O1O2O〉, 〈O†1O
†2O
S†〉)L
≡∫
2<1x≈1,2
ddx1ddx2d
dxDd−2z
vol(SO(d, 2))〈O1(x1)O2(x2)O(x, z)〉〈O†1(x1)O†2(x2)OS†(x, z)〉. (E.10)
84A fixed power of 2 also goes into what we mean by vol(SO(1, 1)).
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JHEP11(2018)102
Here, finite-dimensional Lorentz indices are implicitly contracted. Note that due to the
fixed causal relationships between the points the continuous-spin structures are single-
valued without iε prescriptions (see appendix A). As in the Euclidean case, Lorentzian
three-point pairings are simple to compute because they don’t involve any actual integrals
over positions. We can use the conformal group to fix all three points to a standard
configuration consistent with the given causal relationships, for example
x2 = 0, x1 = e0, x =∞, (E.11)
where e0 is a unit vector in the t direction. The Fadeev-Popov determinant associated with
this choice is 2−d. All that remains is an integral over the polarization vector z,
=1
2dvol(SO(d− 1))
∫Dd−2z 〈O1(e0)O2(0)O(∞, z)〉〈O†1(e0)O†2(0)OS†(∞, z)〉, (E.12)
where vol(SO(d−1)) is the volume of the stabilizer group of the three points.85 In practice,
we can avoid doing the integral over z as well. This is because the product in the integrand
must be proportional to a three-point function of two scalars with dimension d and a
spinning operator with dimension d and spin 2 − d. The integral of the z-dependent part
of this product is always
1
2dvol(SO(d− 1))
∫Dd−2(−2z · e0)2−d =
1
22d−2vol(SO(d− 2)). (E.13)
Thus, we can write(〈O1O2O〉, 〈O†1O
†2O
S†〉)L
=1
22d−2vol(SO(d− 2))
× 〈O1(e0)O2(0)O(∞, z)〉〈O†1(e0)O†2(0)OS†(∞, z)〉(−2z · e0)2−d . (E.14)
F Integral transforms, weight-shifting operators and integration by parts
In this appendix we elaborate on the interplay between integral transforms, weight-shifting
operators, and conformally-invariant pairings, following [68] and generalizing the discussion
to Lorentzian signature. For simplicity of discussion, we ignore possible signs coming from
odd permutations of fermions.
F.1 Euclidean signature
In Euclidean signature we have one integral transform, SE , and a conformally-invariant
pairing
(O, O†) ≡∫ddxO(x)O†(x), (F.1)
85Note that the stabilizer group depends on the causal relationships of the points. For example, three
spacelike points have stabilizer group SO(d− 2, 1).
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JHEP11(2018)102
where the spin indices are implicitly contracted. With respect to this paring we can define
a conjugation on weight-shifting operators and on the integral transform,
(DO, O†) = (O,D∗O†),
(SEO, O†) = (O,S∗EO†). (F.2)
We have ∗2 = 1 and S∗E = SE .
Furthermore, we can define Weyl reflection on weight-shifting operators according to
SED = (SE [D])SE . (F.3)
We then have
S2ED = SE(SE [D])SE = (S2
E [D])S2E , (F.4)
and since S2E = N (∆, ρ), we have when acting on operators transforming in (∆, ρ)
S2E [D] =
N (∆ + δ∆, ρ+ δρ)
N (∆, ρ)D, (F.5)
where (δ∆, δρ) is the weight by which D shifts. Conjugating (F.3) we find
SE(SE [D])∗ = D∗SE , (F.6)
and thus
SE [D]∗ = S−1E [D∗]. (F.7)
We also note that the crossing equation for weight-shifting operators acting on a two-
point function [57] can be written in terms of shadow transform and conjugation. Namely,
we can interpret SED∗ as convolution with the kernel
〈O(DO†)〉, (F.8)
while, on the other hand, it is equal to SE [D∗]S which is convolution with (assume that
DO† transforms as O′†)
〈(SE [D∗]O′)O′†〉. (F.9)
We thus find the crossing equation
〈O(DO†)〉 = 〈(SE [D∗]O′)O′†〉. (F.10)
F.2 Lorentzian signature
The above discussion has an analogue in Lorentzian signature. Now we have more integral
transforms, so let us denote a generic one by W. We also have a new pairing, given by
(O,OS†)L =
∫ddxDd−2zO(x, z)OS†(x, z), (F.11)
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JHEP11(2018)102
where the SO(d − 2) indices are implicit and contracted. This pairing leads to a new
conjugation operation on weight-shifting operators and on integral transforms,
(DO, O†)L = (O,DO†)L,
(WO, O†)L = (O,WO†)L. (F.12)
Note that in general the Lorentzian and Euclidean conjugations do not commute (see
below). Analogously to the Euclidean case, we find
W[D] = W−1[D]. (F.13)
As in Euclidean signature, we can define the action of integral transforms on weight-
shifting operators by
WD = (W[D])W. (F.14)
In principle W[D] can be a differential operator with coefficients which depend on T .
However, when acting on a function, the left hand side of this expression depends only on
the values of this function in a set which fits in one Poincare patch. If W[D] had non-trivial
t dependence, the same would not hold for the right hand side. Therefore W[D] has to be
a local weight-shifting differential operator.
It is easy to check that if two integral transforms commute, then their actions on
weight-shifting operators also commute. Similarly to Euclidean case, relations such as
L2 = fL(∆, J, T ) generalize to action on weight-shifting operators. Let us write down the
square of an order two transform (any transform except R and R)
W2[D] = fW (∆, ρ, T )Df−1W (∆, ρ, T ), (F.15)
where ∆ and ρ are understood as operators which read off the scaling dimension and
representation of whatever they act on. Let us comment on this formula in the case of
S∆. Modulo Wick rotation, we have the relation SE = (−2)JS∆ for traceless-symmetric
operators. It follows that (F.5) and (F.15) should be compatible. That is, we should have
N (∆ + δ∆, J + δJ)
N (∆, J)=
4J+δJf∆(∆ + δ∆, J + δJ , cT )
4Jf∆(∆, J, T ), (F.16)
where δ∆, δJ are the weights by which D shifts, and c is defined by
T DT −1 = cD. (F.17)
I.e. c is the eigenvalue of T in the finite-dimensional irrep of conformal group to which Dis associated. For example, for vector representation c = −1. To check this relation, we
can use the results of section 2.7 and in particular the relation (2.87) which implies (we
consider traceless-symmetric case for simplicity)
f∆(∆, J, T ) = −T −2fL(∆, ρ, T )fJ(1−∆)fL(1− J, 1− d+ ∆, T ). (F.18)
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JHEP11(2018)102
It is then an easy exercise to verify that (F.16) holds for vector weight-shifting opera-
tors [57].
Another useful result is obtained by substituting D →W−1[D] into (F.15) to find
W−1[D] = f−1W (∆, ρ, T )W[D]fW (∆, ρ, T ). (F.19)
For example,
L−1[D] = L[D]fL(∆, ρ, T )
fL(∆ + L[δ∆], ρ+ L[δρ], cT ), (F.20)
where we kept explicit dependence of fL on t, (L[δ∆],L[δρ]) is the weight by which L[D]
shifts. It is easy to check that T -dependence indeed cancels out for D in vector represen-
tation.
We can derive two-point crossing in terms of Lorentzian conjugation and S transform,
〈OS(DOS†)〉 = 〈(S[D]O′S)O′S†〉. (F.21)
Comparing to the Euclidean form of two-point crossing leads to a useful relation
SE [D∗] = S[D]. (F.22)
We will need a version of this relation with order of integral transforms and conjugations
interchanged. First, (F.22) implies
(S−1E [D])∗ = S−1[D]. (F.23)
Then we use that SE and S are proportional to their inverses. In particular, we find
from (F.19)
(f−1E (∆, ρ, T )SE [D]fE(∆, ρ, T ))∗ = (f−1
S (∆, ρ, T )S[D]fS(∆, ρ, T )),
fE(∆, ρ, T )(SE [D])∗f−1E (∆, ρ, T ) = fS(∆, ρ, T )S[D]f−1
S (∆, ρ, T ), (F.24)
where we temporarily interpret SE as a Lorentzian transform defined by (−2)JS∆. We can
now use
fS(∆, ρ, T ) = S2 = S2∆S2
J = 4−JS2ES2
J = 4−JfE(∆, ρ, T )fJ(ρ) (F.25)
to conclude
S[D] = 4Jf−1J (ρ)(S∆[D])∗4−JfJ(ρ). (F.26)
G Proof of (4.48) for seed blocks
In this appendix we prove (4.48) for seed blocks by starting from the scalar case. For
simplicity we consider only bosonic representations. We assume that Oi are in SO(d)
representations appropriate for the seed block for intermediate ρ which we are interested
– 90 –
JHEP11(2018)102
in. As discussed in section 4.4 of [57], we can assume that O2 and O4 are scalars in all
seed blocks, so we don’t have to change their representations. We start with the identity
(〈O†O〉, 〈O†O〉)E(〈O1O2SE [O†]〉)−1E = (〈O′†O′〉, 〈O′†O′〉)ED1,ADA(〈O1O′2SE [O′†]〉)−1
E ,
(G.1)
where D and D are some weight-shifting operators,86 while O′1 and O′ come from a seed
block for which we already know that (4.48) holds. A possible proportionality coefficient
can be absorbed into the definition of either the weight-shifting operators or the tensor
structures. Consider pairing both sides with 〈O1O2SE [O†]〉 to obtain
(〈O†O〉, 〈O†O〉)E(〈O′†O′〉, 〈O′†O′〉)E
= (〈O1O2SE [O†]〉,D1,ADA(〈O1O′2SE [O′†]〉)−1E )E . (G.2)
Integrating by parts and using definitions of appendix F we find
(〈O†O〉, 〈O†O〉)E(〈O′†O′〉, 〈O′†O′〉)E
= (〈D∗1,AO1O2SE [S−1E [D∗]AO†]〉, (〈O1O′2SE [O′†]〉)−1
E )E , (G.3)
which allows us to conclude
〈D∗1,AO1O2SE [S−1E [D∗]AO†]〉 =
(〈O†O〉, 〈O†O〉)E(〈O′†O′〉, 〈O′†O′〉)E
〈O1O′2SE [O′†]〉, (G.4)
or, canceling SE on both sides,
〈D∗1,AO1O2(S−1E [D∗]AO†)〉 =
(〈O†O〉, 〈O†O〉)E(〈O′†O′〉, 〈O′†O′〉)E
〈O1O′2O′†〉. (G.5)
We will use this characterization of D and D later in the proof.
For now, let us apply (G.1) to (4.47) and find that H is given by
H∆,ρ(xi) = −µ(∆, ρ′†)(O1O′2SE [O′†])(〈O1O′2O′†〉, 〈O†1O′†2 O′〉)−1
E ×
×∫
2<x<1ddxDd−2z〈0|D1,AO†1L[DAO](x, z)O′†
2+ |0〉(〈0|O4+L[O](x, z)O3|0〉)−1L .
(G.6)
We now use
L[DAO] = L[D]AL[O], (G.7)
and integrate L[D] by parts. This gives
H∆,ρ(xi) =−µ(∆,ρ′†)(O1O′2S∆[O′†])(〈O1O′2O′†〉,〈O†1O′†2 O′〉)−1
E ×
×∫
2<x<1ddxDd−2z〈0|D1,AO†1L[O](x,z)O′†
2+ |0〉L[D]A
(〈0|O4+L[O](x,z)O3|0〉)−1L ,
(G.8)
86Here tilde isn’t related to shadow transform and D acts on the third position. The representation of
index A can be assumed to be vector.
– 91 –
JHEP11(2018)102
where L[D] acts on the middle position in the right three-point structure. We can further
apply a crossing transformation on the right three-point structure as in [57] to make all
differential operators act on the external operators only. We will not do this in detail, be-
cause we will anyway reverse this step in a moment. Let us denote the resulting differential
operator acting on external operators by D.
The conclusion of the above calculation is schematically that
Hρ = DHρ′ , (G.9)
where Hρ′ is some conformal for which we know (4.48) to hold. We can thus apply D
to (4.48) written for Hρ′ . Since the right three-point structure in (4.48) and (4.47) is the
same, we can unwind the steps in the derivation of D which were performed solely on the
right three-point structure to conclude
H∆,ρ(xi) = − 1
2πi
D1,A
(T2〈O1O′2L[O′†]〉
)−1
LL[D]
A(T4〈O4O3L[O]〉)−1
L
(〈L[O′]L[O′]〉)−1L
. (G.10)
We can use (H.25) to write this as
H∆,ρ(xi) = − 1
2πi
(〈L[O]L[O]〉)−1L
(〈L[O′]L[O′]〉)−1L
S[L[D]]AD1,A
(T2〈O1O′2L[O′†]〉
)−1
L(T4〈O4O3L[O]〉)−1
L
(〈L[O]L[O]〉)−1L
.
(G.11)
We now want to express
S[L[D]]AD1,A(〈O1O′2L[O′†]〉)−1L (G.12)
in terms of
(〈O1O2L[O†]〉)−1L . (G.13)
To do this, let us consider the Lorentzian pairing(〈O1O2L[O†]〉,S[L[D]]AD1,A(〈O1O′2L[O′†]〉)−1
L
)L
=
(S[L[D]]
AD∗1,A〈O1O2L[O†]〉, (〈O1O′2L[O′†]〉)−1
L
)L
. (G.14)
We can use the results of appendix F and 2.7 to write
S[L[D]] = L[S[D]] = L−1[S[D]] =fL(L[∆],L[ρ†], T )
fL(L[∆] + L[δ∆],L[ρ†] + L[δρ], cT )L[S[D]], (G.15)
where (δ∆, δρ) is the weight by which S[D] shifts and c is defined by (F.17) for D. Since we
consider only bosonic representations, c = ±1 (c = −1 for vector weight-shifting operators).
We have (∆ + δ∆, ρ† + δρ) = (∆′, ρ′†). We furthermore have
L[S[D]]L[O†] = L[S[D]O†] =4−JfJ(ρ†)
4−J ′fJ(ρ′†)L[(S∆[D])∗O†] (G.16)
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JHEP11(2018)102
and thus
S[L[D]]AD∗1,A〈O1O2L[O†]〉 =
4−JfJ(ρ†)
4−J ′fJ(ρ′†)
fL(L[∆],L[ρ†], T )
fL(L[∆′],L[ρ′†], cT )〈O1D∗1,AO2L[(SE [D])∗O†]〉.
(G.17)
Now use (SE [D])∗ = S−1E [D∗], apply L to both sides of (G.5) and conclude
S[L[D]]AD∗1,A〈O1O2L[O†]〉= 4−JfJ(ρ†)
4−J′fJ(ρ′†)
fL(L[∆],L[ρ†],T )
fL(L[∆′],L[ρ′†], cT )
(〈O†O〉,〈O†O〉)E(〈O′†O′〉,〈O′†O′〉)E
〈O1O′2L[O′†]〉.
(G.18)
This implies that the pairing (G.14) is equal to
4−JfJ(ρ†)
4−J ′fJ(ρ′†)
fL(L[∆],L[ρ†], T )
fL(L[∆′],L[ρ′†], cT )
(〈O†O〉, 〈O†O〉)E(〈O′†O′〉, 〈O′†O′〉)E
(G.19)
and thus
S[L[D]]AD1,A〈O1O′2L[O′†]〉−1
=4−JfJ(ρ†)
4−J ′fJ(ρ′†)
fL(L[∆],L[ρ†], T )
fL(L[∆′],L[ρ′†], cT )
(〈O†O〉, 〈O†O〉)E(〈O′†O′〉, 〈O′†O′〉)E
(〈O1O2L[O†]〉)−1L . (G.20)
Collecting all the pieces, we find that (G.11) implies (4.48) for the seed H if
C =(〈L[O]L[O]〉)−1
L
(〈L[O′]L[O′]〉)−1L
4−JfJ(ρ†)
4−J ′fJ(ρ′†)
fL(L[∆],L[ρ†], T )
fL(L[∆′],L[ρ′†], cT )
(〈O†O〉, 〈O†O〉)E(〈O′†O′〉, 〈O′†O′〉)E
= 1. (G.21)
Proof that C = 1. First, we note that
(〈O†O〉, 〈O†O〉)E(〈O′†O′〉, 〈O′†O′〉)E
=4J dim ρ†
4J ′ dim ρ′†. (G.22)
Furthermore, fJ is square of shadow transform in d − 2 dimensions. Thus if we write
ρ† = (J, λ) then (similarly to appendix C)
fJ(ρ†) ∝ dimλ
µ(ρ†), (G.23)
where µ is the Plancherel measure for SO(d− 1, 1). Furthermore, the ratio
µ(ρ†)
dim ρ†(G.24)
is independent of ρ [42, 68]. This implies that
4−JfJ(ρ†)
4−J ′fJ(ρ′†)
(〈O†O〉, 〈O†O〉)E(〈O′†O′〉, 〈O′†O′〉)E
=dimλ
dimλ′. (G.25)
– 93 –
JHEP11(2018)102
Furthermore, we can write
dimλ
dimλ′=
(〈O′O′†〉)−1L
(〈OO†〉)−1L
, (G.26)
which is due to
(〈OO†〉, 〈OSOS†〉)L ∝ dimλ, (G.27)
and similarly for primed quantities (see appendix E).
Now we need to recall the calculation of 〈L[O]L[O†]〉. We have for the kernel which is
represented by the time-ordered two-point function 〈OO†〉,
〈OO†〉 = S(1 +
∞∑n=1
γ−n(T n + T −n)), (G.28)
where γ is the eigenvalue of T corresponding to O, see (2.15). The calculation in sec-
tion 4.1.4 then yields, in the same sense as above,
〈L[O]L[O†]〉 = S(1 +∞∑n=1
γ−n(T n + T −n))T −1fL(F[∆],F[ρ], T ). (G.29)
Since L commutes with S, we find that we can replace fL(F[∆],F[ρ],T ) by fL(L[∆],L[ρ],T ).
This implies
(〈L[O]L[O]〉)−1L
(〈L[O′]L[O′]〉)−1L
=(1 +
∑∞n=1 γ
′−n(T n + T −n))fL(L[∆′],L[ρ′], T )
(1 +∑∞
n=1 γ−n(T n + T −n))fL(L[∆],L[ρ], T )
(〈OO†〉)−1L
(〈O′O′†〉)−1L
. (G.30)
Recall that S[D] takes O to O′ and cD = T DT −1, which implies γ′ = cγ = ±g. (Recall
we consider only bosonic representations.) Thus we have
(1 +∑∞
n=1 γ′−n(T n + T −n))
(1 +∑∞
n=1 γ−n(T n + T −n))
fL(L[∆′],L[ρ′], T )
=(1 +
∑∞n=1 γ
′−n(T n + T −n))
(1 +∑∞
n=1 γ′−n((cT )n + (cT )−n))
fL(L[∆′],L[ρ′], T )
=(cT − γ)(cT − γ−1)
(T − γ)(T − γ−1)fL(L[∆′],L[ρ′], T )
= fL(L[∆′],L[ρ′], cT ), (G.31)
where we used the fact that (2.80) is T -independent. We thus conclude that
(〈L[O]L[O†]〉)−1L
(〈L[O′]L[O′†]〉)−1L
=fL(L[∆′],L[ρ′], cT )
fL(L[∆],L[ρ], T )
(〈OO†〉)−1L
(〈O′O′†〉)−1L
. (G.32)
By combining this equation with (G.25) and (G.26) we see that indeed87
C = 1. (G.33)87Since we for simplicity restricted to bosonic representations, we haven’t been very careful with distin-
guishing ρ and ρ†. (There is no difference except possibly for self-dual tensors.) It would be interesting to
repeat our argument in a more careful manner, accounting for fermionic representations as well.
– 94 –
JHEP11(2018)102
H Conformal blocks with continuous spin
H.1 Gluing three-point structures
Consider two three-point structures 〈O1O2O〉 and 〈OO3O4〉. We can glue them into a
conformal block as follows. We find a linear operator B12O(x12) such that in the OPE
limit 1→ 2, the first three-point structure becomes
〈O1O2O†(x)〉 ∼ B12O(x12)〈O(x2)O†(x)〉, (|x12| |x1 − x|, |x2 − x|). (H.1)
For example, when O1,O2,O are all scalars, we have
B12O(x12) = x∆O−∆1−∆212 . (H.2)
(B12O can be extended to a differential operator such that (H.1) becomes an equality away
from the 1 → 2 limit, but this is not necessary for the current discussion.) Note that to
define B12O we must choose a normalization of the two-point structure 〈OO〉.We define a conformal block GOiO (xi) as the conformally-invariant solution to the con-
formal Casimir equation [108] whose OPE limit is
GOiO (xi) ∼ B12O(x12)〈O(x2)O3O4〉, (|x12| |xij |). (H.3)
It is very useful to introduce the following notation for a conformal block, which makes
manifest the choices of two- and three-point structures needed to define it
GOiO (xi) =〈O1O2O†〉〈OO3O4〉
〈OO†〉. (H.4)
In our convention O appears in the OPE O1 ×O2 and O† in the OPE O3 ×O4.
H.1.1 Example: integer spin in Euclidean signature
As an example, let us review the case of external scalars φ1, . . . , φ4 and an exchanged
operator O with integer spin J ,
G∆i∆,J(xi) =
〈φ1φ2O〉〈φ3φ4O〉〈OO〉
, (H.5)
where 〈φ1φ2O〉 and 〈φ3φ4O〉 are the standard three-point structures (A.25) and 〈OO〉 is
the standard two-point structure (A.24). We will assume that all points are in Euclidean
signature.
In the OPE limit 1→ 2, we have
〈φ1φ2O(x0, z)〉 ∼ 1
x∆1+∆2−∆+J12
(−2z · I(x20) · x12)J
x2∆20
=1
x∆1+∆2−∆+J12
xµ112 · · ·x
µJ12 〈Oµ1···µJ (x2)O(x0, z)〉. (H.6)
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JHEP11(2018)102
To compute the leading behavior of the block, it suffices to take the limit 3 → 4 in 〈φ3φ4O〉,
〈φ3φ4Oµ1···µJ (x2)〉 =1
x∆3+∆4−∆+J34
(−2I(x42) · x34)µ1 · · · (−2I(x42) · x34)µJ − traces
x2∆42
.
(H.7)
(This limit is identical to the first line of (H.6) after replacing 1, 2, 0→ 3, 4, 2 and stripping
off the polarization vector z.) Thus the OPE limit of the resulting block is
G∆i∆,J(xi) ∼
xµ112 · · ·x
µJ12
x∆1+∆2−∆+J12 x∆3+∆4−∆+J
34
(−2I(x42) · x34)µ1 · · · (−2I(x42) · x34)µJ − traces
x2∆42
=1
x∆1+∆212 x∆3+∆4
34
(x2
12x234
x442
)∆/2
2J CJ
(−x12 · I(x42) · x34
|x12||x34|
). (H.8)
Here, we’ve used the identity
(mµ1 · · ·mµJ )(nµ1 · · ·nµJ − traces) = |m|J |n|J CJ(m · n|m||n|
), (H.9)
where
CJ(η) =Γ(d−2
2 )Γ(J + d− 2)
2JΓ(J + d−22 )Γ(d− 2)
2F1
(−J, J + d− 2,
d− 1
2,1− η
2
)(H.10)
is proportional to a Gegenbauer polynomial (note in particular that for η = 1 the hy-
pergeometric function reduces to 1). Factoring out some standard kinematical factors,
we find
G∆i∆,J(xi) =
1
(x212)
∆1+∆22 (x2
34)∆3+∆4
2
(x2
14
x224
)∆2−∆12
(x2
14
x213
)∆3−∆42
G∆i∆,J(χ, χ), (H.11)
where G∆i∆,J(χ, χ) is a solution to the conformal Casimir equations normalized so that
G∆i∆,J(χ, χ) ∼ (χχ)∆/2
(χ
χ
)−J/2, (χ χ 1). (H.12)
Here, χ, χ are conformal cross-ratios defined by u = χχ, v = (1 − χ)(1 − χ). This is the
standard conformal block in the normalization convention of [16, 31].
H.1.2 Example: continuous spin in Lorentzian signature
Our definition of a conformal block also works when O has continuous spin. However, now
we must allow B12O to be an integral operator in the polarization vector of O. Let us
again consider external scalars φ1, . . . , φ4. For later applications, we work in a Lorentzian
configuration where all four points 1, 2, 3, 4 are in the same Minkowski patch, with the
causal relationships 1 > 2, 3 > 4, and all other pairs spacelike-separated, see figure 16.
– 96 –
JHEP11(2018)102
1
2
3
4
Figure 16. A configuration of points where 1 > 2 and 3 > 4, with all other pairs of points
spacelike-separated. The three-point structure (H.13) is positive in this configuration.
We also modify the three-point structures by taking x234 → −x2
34 and x212 → −x2
12 so
that they are positive when x0 is spacelike from 1, 2 and 3, 4, since precisely these positive
structures will appear later. Specifically, let
T∆1,∆2
∆,J (x1, x2, x0, z) =(2z · x20x
210 − 2z · x10x
220)J
(−x212)
∆1+∆2−∆+J2 (x2
10)∆1+∆−∆2+J
2 (x220)
∆2+∆−∆1+J2
. (H.13)
We will study the block
T∆1,∆2
∆,J T∆3,∆4
∆,J
〈OO〉, (H.14)
where 〈OO〉 is the two-point structure (A.24). To define a block, our structures only need
to be defined when x0 is spacelike from the other points, so we do not need to give an iε
prescription here.
In the OPE limit 1→ 2, we have
T∆1,∆2
∆,J (x1, x2, x0, z) ∼ 1
(−x212)
∆1+∆2−∆+J2
(−2z · I(x20) · x12)J
(x220)∆
(1→ 2). (H.15)
The quantity on the right differs from the two-point structure 〈O(x2, z′)O(x0, z)〉 by the
replacement z′ → x12. We can no longer strip off z′ and contract indices with x12. However,
the replacement can be achieved via an integral transform:
T∆1,∆2
∆,J (x1, x2, x0, z) ∼ B12O〈O(x2, z′)O(x0, z)〉
B12Of(x′, z′) =1
(−x212)
∆1+∆2−∆−J−d+22
Γ(J + d− 2)
πd−2
2 Γ(J + d−22 )
×∫Dd−2z′(−2x12 · z′)2−d−Jf(x′, z′). (H.16)
– 97 –
JHEP11(2018)102
Now let us apply B12O to the three-point structure T∆3,∆4
∆,J (x3, x4, x2, z), working in
the limit 3 → 4 (since this is sufficient to determine the small cross-ratio dependence of
the resulting block). In doing so, we need the identity∫Dd−2z′ (−2x12 · z′)2−d−J(−2z′ · I(x42) · x34)J
= (−x212)
2−d−J2 (−x2
34)J2
22−dvol(Sd−2)
CJ(1)CJ
(−x12 · I(x42) · x34
(−x212)1/2(−x2
34)1/2
), (H.17)
where CJ(η) is given in (H.10). (Here, it is important that we use the correct definition of
CJ for non-integer J .) Using (H.17), we find that in the OPE limit
T∆1,∆2
∆,J T∆3,∆4
∆,J
〈OO〉∼ 1
(−x212)
∆1+∆22 (−x2
34)∆3+∆4
2
(x2
12x234
x442
)∆/2
2J CJ
(−x12 · I(x42) · x34
(−x212)1/2(−x2
34)1/2
),
(H.18)
so that
T∆1,∆2
∆,J T∆3,∆4
∆,J
〈OO〉=
1
(−x212)
∆1+∆22 (−x2
34)∆3+∆4
2
(x2
14
x224
)∆2−∆12
(x2
14
x213
)∆3−∆42
G∆i∆,J(χ, χ).
(H.19)
This is the same result we would have gotten by pretending J was an integer and performing
the computation in the previous subsection. However, here we see that a conformal block
with non-integer J is well-defined and completely specified by continuous-spin two- and
three-point structures.
H.1.3 Rules for weight-shifting operators
Let us consider how the gluing rule described in H.1 interacts with weight-shifting operators
changing the internal representation. Suppose we can write
〈O1O2O†(x)〉 = 〈O1(DAO′2)(DAO′†)〉 (H.20)
for a pair of weight-shifting operators D and D. By acting with the same weight-shifting
operators on (H.1) for primed operators we find
〈O1O2O†(x)〉 ∼ (D2,AB12O)(x12)〈O(x2)(DAO′†)(x)〉. (H.21)
Recall the crossing equation (F.21), which holds when the two-point structures are related
to the kernel of S-transform. Let us assume for now that this is the case. Then we find
〈O1O2O†(x)〉 ∼ (D2,AB12O)(x12)〈(S[D]AO)(x2)O′†(x)〉. (H.22)
Substituting this into (H.3), we find
GOiO (xi) ∼ (D2,AB12O)(x12)〈(S[D]AO)(x2)O3O4〉. (H.23)
– 98 –
JHEP11(2018)102
Using notation (H.4) we can summarize this as88
〈O1(DAO2)(DAO′†)〉〈OO3O4〉〈OO〉
=〈O1(DAO2)O′†〉〈(S[D]AO)O3O4〉
〈O′O′〉. (H.24)
This holds if the two-point functions for O and O′ are standard in the sense of being related
to S-kernel. Generalization of this to arbitrary two-point functions is given by
〈O1(DAO2)(DAO′†)〉〈OO3O4〉〈OO〉
=〈O′O′〉〈OO〉
〈O1(DAO2)O′†〉〈(S[D]AO)O3O4〉〈O′O′〉
, (H.25)
where the ratio of two-point functions is a scalar defined as
〈O′O′〉〈OO〉
≡ 〈O′O′〉
〈O′O′〉0〈OO〉0〈OO〉
, (H.26)
where the structures with subscript 0 are standard and related to S-kernel. Note that we
can reverse (H.25) by replacing D → S−1[D]. However, due to (F.13) we have S−1[D] =
S[D] and so we get the same rule for moving the operator from right to left.
H.2 A Lorentzian integral for a conformal block
Conformal blocks in Euclidean signature can be computed via a “shadow representation,”
where one integrates a product of three-point functions over Euclidean space [33, 34, 109].
However, this integral produces a linear combination of a standard block G∆i∆,J and the
so-called “shadow block” G∆id−∆,J . The shadow block comes from regions of the integral
where the OPE is not valid inside the integrand.
By contrast, there is a simple integral representation for a block alone (without its
shadow) in Lorentzian signature [110]. The reason is that in Lorentzian signature, we can
integrate over a conformally-invariant region that stays away from two of the points, say
x3,4. Thus, the x3 → x4 OPE limit can be taken inside the integrand and dictates the
behavior of the result.
The Lorentzian integral for a conformal block plays an important role in section 4.1.2,
so let us compute it. Consider the same configuration as in the previous subsection where
1, 2, 3, 4 are in the same Poincare patch, with 1 > 2 and 3 > 4, and other pairs of points
spacelike separated from each other (figure 17). We can produce a conformal block in the
1, 2→ 3, 4 channel by performing a shadow-like integral over the causal diamond 2 < 0 < 1,
G∆,J ≡∫
2<0<1ddx0D
d−2z |T∆1,∆2
d−∆,2−d−J(x1, x2, x0, z)|T∆3,∆4
∆,J (x3, x4, x0, z) (H.27)
The notation |T∆1,∆2
d−∆,2−d−J | means that spacetime intervals xij should appear with absolute
values |xij |, so that the integrand is positive in the configuration we are considering. (This
notation is somewhat imprecise, since when ∆1,∆2,∆, J are complex, we do not mean
one should take the absolute value of the whole expression.) When J is an integer, there
88The results of [57] concerning weight-shifting of the internal representation are recovered by further
using crossing for the weight-shifting operator acting on the right three-point structure.
– 99 –
JHEP11(2018)102
1
2
3
4
0
Figure 17. In the Lorentzian integral for a conformal block, the point x0 is integrated over the
diamond 2 < 0 < 1 (yellow). Because the integration region is far away from points 3, 4, the 3× 4
OPE is valid inside the integral.
is a similar integral expression for a Lorentzian block with∫Dd−2z replaced by index
contractions. However (H.27) also works for continuous spin.
The expression (H.27) is proportional to G∆,J because it is a conformally-invariant
solution to the Casimir equation whose OPE limit agrees with the OPE limit of T∆3,∆4
∆,J
(because the integration point stays away from x3,4). The behavior of the integral in the
limit 1 → 2 is not immediately obvious. However, conformal invariance requires that this
limit must be the same as 3→ 4.
More precisely, in the OPE limit 3→ 4, we have
T∆3,∆4
∆,J (x3, x4, x0, z) ∼ B34O〈O(x4, z′)O(x0, z)〉 (3→ 4, 0 ≈ 3, 4), (H.28)
where B34O is the linear operator defined in (H.16). Plugging this in, we find
G∆,J ∼ B34O
∫2<0<1
ddx0Dd−2z |T∆1,∆2
d−∆,2−d−J(x1, x2, x0, z)|〈O(x4, z′)O(x0, z)〉 (3→ 4).
(H.29)
The integral in the OPE limit now takes the form of an S-transform.
H.2.1 Shadow transform in the diamond
Let us evaluate the integral (H.29) by splitting it into two steps: first we apply S∆ and
then subsequently SJ . For notational convenience, define
∆0 ≡ d−∆
J0 ≡ 2− d− J. (H.30)
– 100 –
JHEP11(2018)102
The S∆ transform is fixed by conformal invariance up to a coefficient a∆1,∆2
∆0,J0,
S∆0[|T∆1,∆2
d−∆,2−d−J(x1, x2, x0, z)|θ(2 < 0 < 1)]
=
∫2<0<1
ddx01
x2(d−∆0)04
|T∆1,∆2
d−∆,2−d−J(x1, x2, x0, I(x04)z)|
= a∆1,∆2
∆0,J0
|2z · x14x224 − 2z · x24x
214|J0
|x12|∆1+∆2−(d−∆0)+J0 |x14|∆1+(d−∆0)−∆2+J0 |x24|∆2+(d−∆0)−∆1+J0. (H.31)
Here, we are writing expressions valid in the kinematical configuration we are consider-
ing, namely 2 < 0 < 1 and 4 ≈ 1, 0, 2. To find the coefficient, we choose the following
configuration in lightcone coordinates
x0 = (u, v, x⊥),
x1 = (1, 0, 0),
x2 = (0, 1, 0),
x4 = (∞,∞, 0),
w = I(x04)z = (2, 0, 0), (H.32)
where the metric is x2 = uv + x2⊥. Note that since 4 is sent to infinity, w is actually
independent of x0. Our integral becomes
a∆1,∆2
∆0,J0=
1
2J0+1
∫dudvdx⊥
|2w ·x10x220−2w ·x20x
210|J0
|x12|∆1+∆2−∆0+J0 |x10|∆1+∆0−∆2+J0 |x20|∆2+∆0−∆1+J0
=vol(Sd−3)
2
∫dudvdrrd−3 (u(1−u)−r2)J0
(u(1−v)−r2)∆1−∆2+∆0+J0
2 (v(1−u)−r2)∆2−∆1+∆0+J0
2
.
(H.33)
It is now straightforward to perform the v integral over v ∈ [ r2
1−u ,u−r2
u ], followed by the
r integral over r ∈ [0,√u(1− u)], and finally the u integral over u ∈ [0, 1]. The result is
a∆1,∆2
∆0,J0=π
d−22 Γ(2−∆0)Γ( 2−J0−∆0+∆1−∆2
2)Γ( d+J0−∆0+∆1−∆2
2)Γ( 2−J0−∆0−∆1+∆2
2)Γ( d+J0−∆0−∆1+∆2
2)
2Γ(1+ d2−∆0)Γ(2−J0−∆0)Γ(d+J0−∆0)
.
(H.34)
Note that a∆1,∆2
∆0,J0= a∆1,∆2
∆0,2−d−J0, which is consistent with the requirement that S∆ commute
with SJ . We can additionally perform SJ using∫Dd−2z′(−2z · z′)2−d−J0(−2z′ · v)J0 =
πd−2
2 Γ(−J0 − d−22 )
Γ(−J0)(−v2)
d−22
+J0(−2z · v)2−d−J0 .
(H.35)
Combining everything together, we find
S0[|T∆1,∆2
d−∆,2−d−J(x1, x2, x0, z)|θ(2 < 0 < 1)] = b∆1,∆2
∆,J T∆1,∆2
∆,J (x1, x2, x4, z)
b∆1,∆2
∆,J ≡πd−2
2 Γ(J + d−22 )
Γ(J + d− 2)a∆1,∆2
d−∆,2−d−J . (H.36)
– 101 –
JHEP11(2018)102
Plugging this into (H.29) and using (H.19), we conclude
G∆,J(xi) =b∆1,∆2
∆,J
(−x212)
∆1+∆22 (−x2
34)∆3+∆4
2
(x2
14
x224
)∆2−∆12
(x2
14
x213
)∆3−∆42
G∆i∆,J(χ, χ). (H.37)
H.3 Conformal blocks at large J
In this appendix, we compute the large-J behavior of a conformal block. Recall that we
have the decomposition
G∆i∆,J(χ, χ) = gpure
∆,J (χ, χ) +Γ(J + d− 2)Γ(−J − d−2
2 )
Γ(J + d−22 )Γ(−J)
gpure∆,2−d−J(χ, χ). (H.38)
Thus it suffices to compute the large-J behavior of gpure∆,J .
The Casimir equation was solved in the large-∆ limit in [111, 112]. We can use this
result together with an affine Weyl reflection to determine gpure∆,J at large J . The solution
from [112] is given by
r∆fJ(cos θ)
(1− r2)d−2
2 (1 + r2 + 2r cos θ)12
(1+∆12−∆34)(1 + r2 − 2r cos θ)12
(1+∆34−∆12)(|∆| 1),
(H.39)
where r and θ are defined by
ρ = reiθ, ρ = re−iθ, χ =4ρ
(1 + ρ)2, χ =
4ρ
(1 + ρ)2. (H.40)
From studying the regime r 1, we find that fJ(cos θ) must obey the Gegenbauer differ-
ential equation.
Note that the conformal Casimir equation has the following symmetries:
(∆, J)↔ (1− J, 1−∆),
r ↔ w = eiθ. (H.41)
The first is an affine Weyl reflection that preserves the Casimir eigenvalue. The second
transformation is equivalent to ρ↔ 1/ρ, which leaves χ invariant, and therefore also leaves
the Casimir equation invariant. Applying these transformations to (H.39), we find
w1−Jf1−∆
(12(r+ 1
r ))
(1−w2)d−2
2 (1+w2+w(r+1/r))12
(1+∆12−∆34)(1+w2−w(r+1/r))12
(1+∆34−∆12)(|J | 1).
(H.42)
Note in particular that we have replaced large-∆ with large-J . Demanding pure power
behavior as r → 0 requires us to choose the following solution to the Gegenbauer equation:
fJ(x) = (2x)J2F1
(−J2,
1− J2
, 2− J − d
2,
1
x2
). (H.43)
Finally, fixing the constant out front and rearranging terms, we find (5.13).
– 102 –
JHEP11(2018)102
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